Existence of neutrinos. Basically theory predicted an extra particle being generated in nuclear reactions than were observed, and (IIRC) Wolfgang Pauli was the one who first suggested that a neutral, very weakly interacting and thus difficult to observe particles were being generated in nuclear reactions. Enrico Fermi termed the hypothesized particles as neutrinos; this happened in the 1930's. Detection of neutrinos was first published in 1956, and although neutrinos have had some surprises to offer to particle physics - primarily, neutrino oscillation - they are essentially pretty much like Pauli hypothesized them to be like.
Formation of solar system from accretion disk around the star. Obviously going to past to check out if this applies to our solar system is right out, but with increasingly powerful telescopes, it's possible to find infantile solar systems in different phases of developement, and the observations seem to in fact suggest that solar systems are born much like hypothesized.
Space-time curvature due to gravity, predicted by general theory of relativity and confirmed with both starlight bending observations during lunar eclipses, gravitational lenses and the travelling perihelion of Mercury's orbit.
Speciation - Darwin pretty much predicted that isolated populations of same species can evolve into separate species, and there have been a number of
observed instances of speciation.
Hygiene - especially medical hygiene; Semmelweis, Lister et al predicted - even without exact knowledge of what was going on - that filthy conditions might have something to do with infections and diseases in general, and it actually worked when it was practiced. Similarly it was predicted that something, whatever you wanna call them, causes diseases and infections, and they aren't in fact the scource of God or somesuch nonsense. And indeed bacteria and viruses and parasites were found when research methods become sophisticated enough to see them.
The (recorded) concept of spherical Earth emerged during the 6th century BCE, and Erastothenes of Cyrene measured the circumference of Earth within 10% error bars. The fact of spherical Earth was ultimately confirmed by Fernão de Magalhães (much like Cristóbal Colón is known as Cristopher Columbus, Magalhães is known as Ferdinand Magellan) when he commited the first recorded sailing trip around the world.
Also, it was long speculated that asteroids and comets might sometimes hit planets, but
proof of this was only recorded in large scale when Jupiter was hit by a comet and the event was recorded (it's pretty spectacular footage, too). Dunno if that counts as prediction being confirmed...
The problem with really long term predictions is obviously that while it can be predicted very accurately that, for example, Sun will render Earth uninhabitable in 500-1000 million years, it's difficult to confirm that with experimentation in individual cases because of obvious reasons. However, it is possible to observe similar stars in different stages of their developement, build a general model of stellar life cycle based on observations, and predict within certain error bars that the Sun will grow too hot for water to stay liquid on Earth in 500-1000 million years, and that it will continue brightening until at some point about 4.5-5 billion years in future it will expand rapidly, then contract to white dwarf star, likely after blowing it's outer surface into space to form a planetary nebula.
Other predictions? Well, Moore's law seems to be eerily accurate thus far. And while the transportation and life in general is not quite like they envisioned in the 50's (no moon hotels, plastic bubble houses and flying cars), some aspects of society - like computation and communications - have developed surprisingly unsurprisingly. Broadband video calls? Check. Personal communicator devices? Check. Personal communicator devices with video calls? Yeah, very soon. Computation has developed pretty much as expected after the introduction of Moore's law; previous expectations have actually been largely surpassed.
