I almost don't feel like jumping into the thread soup, but as the founder of a major anime group.... *breaks out boots*
Here's what I would call some "Staples of Anime" or "some things after you watch, you can probably find someone in most any anime group that has also watched or can comment on."
Castle In The Sky (dub)
Gun Buster OVA (sub) + Dai Buster OVA (dub)
Howl's Moving Castle
Lupin III: The Castle Of Cagliostro (dub)
Macross Plus (dub)
Nausicaä Of The Valley Of The Wind (dub)
Princess Mononoke (dub)
Spirted Away (dub)
Summer Wars (sub)
The Place Promised In Our Early Days (dub)
The Wind Rises
Ah! My Goddess + Movie and Sequels (sub preferably, dub ok, OVA is related but it's own thing)
Blue Submarine No.6 (dub)
Boogiepop Phantom (dub)
Cowboy Bebop + Movie (dub)
Eden of the East
Eureka 7 (dub) (also note this series takes 23 episodes to get moving, but it does end amazingly)
Fist Of The North Star
Full Metal Panic + Sequels and Spin Off Series Fumoffu (dub)
Galaxy Express 999 (dub)
G Gundam (dub)
Ghost In The Shell: Stand Alone Complex + 2nd Gig + Solid State Society Movie(dub)
Gundam: 0080 (dub)
Gundam: 0083 (dub)
Gundam: 8th Ms Team (dub)
Gundam Wing + Endless Waltz Movie (dub)
High School Rumble (sub preferably)
Irresponsible Captain Tyler the TV
Last Exile (dub)
Love Hina + the Christmas Movie
Lupin III: Season 2 (dub)
Macross + Many Sequels and Movies
Magical Shopping Arcade Abenobashi
Neon Genesis Evangelion (dub)
One Punch Man
Outlaw Star (most will remember the Toonami version, uncut is rather different)
Pani Poni Dash (or Lucky Star)
Read or Die Movie + TV series
Rurouni Kenshin + (prequel movie Samurai X) (dub)
Serial Experiments Lain (dub)
Star Blazers aka Space Battleship Yamato
Stein's;Gate + OVA + Deja Vu Movie (this is a series that many drop prior to ep 12, it does a lot of world building to get to activate the plot)
Tenchi Muyo OVA 1 + 2
Tenchi Universe + TMiL and Tenchi Forever
The Big O (dub)
The Tower of Druaga S1 & S2
The Galaxy Railways (dub)
X'amd Lost Memories
Yu Yu Hakusho
Stuff coming out of the current season that may be worth a shot:
Quan Zhi Gao Shou (actually Chinese animation, alternative title: SAO 35: Chinese Kirito gets fired and holes up in a netcafe.)
Little Witch Academia (continuing from last season)
Alice to Zouroku
Manga - Japanese comics, naturally in reversed reading order from English (flipped), tend to be thicker and image heavy.
Anime - Japanese produced animation. Koreans and Chinese have been doing work for the Japanese for years and their own studios are starting to take off. In Japan this term refers to any form of animation. Outside of Japan it refers specifically to Japanese animation, and thus is synonymous with the term “Japanimation.”
Manhwa - South Korean comics similar to manga
Even More Terminology!
AMVs: Anime music videos. These are generally amateur blendings of anime clips with unrelated songs. Some are done seriously, while others are gleeful parodies. They are a popular feature at anime conventions.
baka: A term commonly heard while watching subtitled anime, it translates variously as “stupid,” “fool,” or “idiot.”
bento: A traditional Japanese boxed lunch.
bishojo/bishoujo: A “pretty girl.”
bishonen: A “pretty boy.”
chibi: In Japan this term means “small” or “undersized” and is often interpreted as “runt” or “puny.” In the American fan community it has lost its negative connotation and refers to characters that are cute and either tiny or superdeformed.
cosplay: Short for “costume play,” this term refers to the practice of anime fans dressing up as their favorite anime characters and acting out either scenes from the anime or original situations involving the characters. A staple feature of anime conventions on both sides of the Pacific.
dating sim: Short for “dating simulation,” it’s a popular type of interactive first-person computer game in Japan (but virtually unknown in North America beyond the anime fan community) which involves the main character trying to hook up with one or more of a variety of possible romantic interests. Dating sims typically feature multiple possible endings and little or no animation, but sometimes have hidden characters. They are common source material for anime series and sometimes pop up in anime; Gene Starwind in Outlaw Star and Il Palazzo in Excel Saga have both been seen playing them, for instance.
dojinshi/doujinshi: Fan-produced fiction using characters from a popular manga or anime series. Selling it is technically illegal under Japanese copyright laws, although an underground market for it is tolerated as a promotional tool. It is hugely popular in Japan (especially amongst female fans) but has not caught on much in the States.
ecchi (pronounced eh-chee): Means “perverted,” but it has milder connotations than hentai. Ecchi content would be roughly equivalent to a Playboy pictorial by American standards.
ero game/hentai game: An erotic adults-only version of a dating sim where the focus is on having sex with one or more potential candidates. These games usually feature very explicit visuals and can explore very dark themes. (Bondage or some degree of S&M content is common, and games based on rape are not unknown.)
fanboy/fangirl: A generally uncomplimentary reference to an obsessive anime fan in the States. Roughly equivalent to “Trekkie.”
fan service: Although most commonly interpreted to mean nudity or shots of a women’s figure or undergarments that are purely gratuitous, “fan service” can also refer to any elements put into an anime or manga for no other reason than to be fan-pleasing.
fansub: Anime that has been subtitled by an amateur individual or group, usually because professional subtitling and/or dubbing hasn’t been done yet. Most anime releases are available in the U.S. in fansub form months or even years before a professional job is done (if it ever is), often within a week or two of the airing of each individual episode on Japanese TV. They once were mostly seen at conventions and in college anime groups, but distribution over the Internet via YouTube, IRC, or p2p resources like BitTorrent has become the common practice in the 2000s. Parody fansubs of popular titles also do exist. Though fansubs violate copyright laws, they are usually (but not always) tolerated as a means of promoting interest in a title prior to its official translation and release, provided that they are not sold and cease when a title gets officially licensed. Many (but not all) Internet sites that allow downloading or file-swapping of fansubs will pull titles from circulation once they have officially been licensed for U.S. distribution.
graphic novel (GN): Several issues of comic books or manga collected into a single book-like format, or (less commonly) an original story or novel adaptation presented in a comic book format that is bound in book form. Manga graphic novels are usually the size of an extra-wide paperback novel
harem series: A popular subgenre of romantic comedy which centers on a single boy/young man surrounded by a bevy of young beauties who cohabitate with him. These series usually take on one of two basic forms: either the male lead is romantically attached to one of the beauties but is tempted by others, or the male lead is indecisive and most of the female characters are vying for his attention. Typically the male lead is a kind and gentle soul while most of the female characters are some combination of more powerful, aggressive, and/or capable.
hentai: Literally translating as “perverted,” when used in reference to an anime type it refers to adults-only anime distinguished by explicit sexual content. Most hentai would be considered porn by American standards. Be forewarned that some hentai titles go to great extremes; rape scenes, S&M content, and “tentacle monsters” (think about it) are not uncommon, for instance.
hikikomori: An extreme antisocial phenomenon in Japan (though it has parallels in other postindustrial countries) which involves individuals completely withdrawing from work, school, family, and society in general and holing up in their rooms, often spending all their time playing games, reading manga, watching anime, and/or surfing the Web. It has recently appeared in several anime and manga, among them Hayate the Combat Butler and Welcome to the NHK!
honorifics: The English language uses titles – Mr., Mrs., Lord, Sir, etc. – in front of names in formal situations. Honorifics in Japanese, which are tagged on to the end of a name, serve a similar purpose but are also used to denote the level of formality and familiarity which a speaker is using in addressing the named person. Failure to use an honorific is either a show of contempt for the subject or a sign of intimate familiarity with the subject – which can, of course, be just as insulting if used improperly. Honorifics have traditionally either disappeared or been replaced by English equivalents during professional translations, but lately their partial or complete retention in both subtitling and English dubs has become more common.
Honorifics often heard in anime include:
-chan: Expresses intimacy and/or affection for a close friend or relative, especially a female one. May also be used toward a child or pet.
-dono: Honorable, often translated as “Sir” or “Lord.” This is an archaic term not used much in regular conversation.
-kun: Used in addressing male colleagues or students. (May be used for women, but this is not common.)
-nii/nee: Big brother/big sister
-sama: Mr., Mrs., etc. Used in the most formal or respectful sense. Not commonly used in regular conversion.
-san: Mr., Mrs., etc. Used in cases where the subject is of roughly equal social standing and formality is preferred (such as when greeting a stranger).
-senpai: Senior, elder, or upperclassmen. Used in cases where the subject is a superior within the same social class.
josei: A category of manga (and anime) aimed at older (18+) female readers and usually featuring adult female characters. Tends to emphasize everyday experiences and realistic (rather than idealistic) romance. It is also sometimes used within anime and manga to refer to a sexual preference for older women.
kana: The phonetic version of written Japanese. Divided into two types: hiragana, which is used to modify kanji and write Japanese and Chinese-borrowed words phonetically, and katakana, which is most commonly used to write words and names borrowed from languages other than Chinese but can also be used for particular emphasis (roughly equivalent to putting a word in italics or bold print in English). Both use 46 basic characters which can be combined to represent one hundred different syllables.
kanji: Chinese characters that compose one of the three basic scripts used for written Japanese (hiragana and katakana being the other two). Kanji characters represent either an entire word or the stem of a word depending on usage. Though the symbols are more complex and not conducive to phonetic writing (and hence to being used for words borrowed from non-Chinese languages), their meaning is often more clear than the phonetically-written versions of the same word - an important consideration in a language heavily prone to homonyms.
kawaii: Along with baka, this is one of the most recognizable words in the original Japanese vocal tracks for American fans who don’t speak Japanese. It translates as “it’s so cute” and is often said by female characters with great emphasis.
lolicon: An English transliteration of the Japanese abbreviation for “Lolita Complex.” It refers to sexual/erotic imagery of underage girls (or those who look underage) in anime/manga.
mahou shojo/magical girl: The latter is just the English translation of the former. Both refer to a genre targeted at teen and preteen girls. Such titles always feature one or more (generally middle school-aged) girls who can transform into alter egos capable of working powerful magic. Elaborate transformation scenes, typically replayed at least once each episode, and massive cutesiness are hallmarks of the genre.
mecha: A genre of anime that prominently incorporates giant robots and/or combat suits. Can also refer to the robots/combat suits themselves.
moe or moé: A characteristic of female anime/manga characters which inspires a protective and/or loving response from the audience. (Note that helplessness is not a prerequisite, as some very powerful girls, such as Chise from Saikano, can also be considered moe.) Most commonly such characters are either physically or mentally young (i.e. naïve or innocent in outlook) and have some obvious sympathetic weakness that they strive to correct. Moe is implicitly non-sexual in connotation (unlike lolicon) but is sometimes sexualized after the fact.
omake: Bonus animation included on a videotape or DVD; a close American equivalent would be the animated shorts once commonly used to front movies. This bonus animation often features superdeformed characters.
otaku: A dedicated fan. In Japan this term implies unhealthy obsessiveness with some hobby, while in the U.S. it carries no pejorative connotation and hence is used proudly by anime fans.
OVA/OAV: Original Video Animation, aka straight‑to‑video. Sometimes these productions are standalone series, other times they are used to test the market to see if the story is popular enough to warrant a TV series. They are also sometimes used to round out a TV series by providing “bonus” episodes containing content which, due to time constraints or its graphic nature, could not be aired on TV.
OST: Original Soundtrack.
reverse harem series: A variation on the standard harem series which involves one girl cohabitating with several handsome guys, some or all of whom are potential romantic interests. Fruits Basket is probably the best-known example, but there are others.
romanization: Translation of Japanese writing into an alphabet system.
romaji: Japanese written with a European-based alphabet.
ronin: Traditionally this term refers to a masterless samurai. In recent times it has also come to refer to college-age students who are attending cram schools because they have not yet passed college entrance exams.
seinen: A category of manga (and anime) targeted at adult males. Can include pornographic titles but does not automatically have such content. Equivalent to Adult or Mature Audiences titles in the States.
seiyuu: A Japanese voice actor.
senpai (sometimes also spelled sempai): A title given to an upperclassman or other figure senior to the speaker.
shojo/shuojo: Technically this refers only to anime based on manga from girls’ magazines. It is more commonly interpreted in the U.S. as referring to anime and manga specifically targeted at girls.
shojo ai: Literally translates as “girl love” and refers to stories about girl-girl couples that stress romantic (rather than erotic) elements. It is nowhere near as commonly-used as shonen ai or yuri and is rare to find in the States.
shonen ai/shuonen ai: Literally translates as “boy love” and refers to stories about male couples. Although its use is out of fashion in Japan, it is generally used in the U.S. to refer to stories about male homosexual relationships that stress romantic (rather than erotic) elements.
superdeformed (SD): A miniaturized and caricatured rendition of an anime character. Most commonly seen in omake and comedy anime. There is no parallel in American animation.
tsundere: A personality that is initially combative but becomes loving and emotionally vulnerable. The classic example is Asuka Langley Soryu of Neon Genesis Evangelion.
visual novel: A semi-interactive Japanese computer “game” which tells a story through still visuals and text flashed across the bottom of the screen. Unlike dating sims, they offer minimal choices that usually have little impact on the story. Some have been known to be adapted into anime.
yaoi (pronounced yah-oy): An acronym for a Japanese phrase that provides an uncomplimentary view on gay relationships, this term can refer either to stories involving male homosexual relationships or to fan art and stories pairing male characters from popular series. In the U.S. the term carries more homoerotic connotations.
yuri: The Japanese word for “lily flower,” the term traditionally refers to stories about lesbian relationships written by men for men (although they are also seen in manga targeted at women). In the U.S. the term generally refers to erotic stories featuring lesbians.
Here's a random anime history 101 as to how it all got started around 1907. https://www.youtube.com/watch?v=-XlS6CYVNy0