Okay.
The most basic thing we can assume about the universe is that the laws of nature are same to all observers, regardless of their state of motion.
The speed of light (actually speed of all electromagnetic wave motion) arises from the structure of empty space. More specifically, in Maxwell's equations it is shown that the speed of light depends only about the permeability and permittivity of the vacuum. Also, since there is no universal base co-ordinates to reference in a vacuum, that indeed means that everyone will observe the speed of light in vacuum to be constant, c, since vacuum itself will appear exactly the same to everyone passing through it - you can't reference to nothingness!
Now, when a situation like what you described occurs - an observer (A) sends a light beam (or a photon) forwards while moving at velocity v in relation to another observer (B) - we can from the previous deduction simply say that both observers will see that the photon moves at the constant speed of light, c. This is a fact, and all that follows has variable interpretations and requires a bit deeper analysis to understand.
Because there is a speed difference, yet both must observe the speed of light to be constant, we can deduct some interesting things. Mathematically, you can derive the so-called Lorenz-contractions for distance and time for both observers, based on the speed difference. It all gets a bit hairy when it's written with a computer so I won't bother to start with (I'll do a copy-paste later if it's necessary). The basic idea is that when relative velocity increases, the observers start seeing things different from each other. If both were stationary, they would see all distances similarly, and time would pass at similar rate for both observers. However, if there's a large speed difference between observers, the following (and some) happens:
-when a stationary observer observes a moving target which has a clock on it's surface, he will observe that the clock is running slower than his own clock. Note that this applies to both observers - they both see the other's time is running slower than their own. It is a bit difficult to explain, but there's no paradoxes there in the end.
-same applies to the length on the direction of velocity. A stationary observer will note that a fast-moving observers' one-metre long scale looks somewhat shorter. This can similarly be inverted, since both observers will consider themselves static observers and the other one to be on the move.
Now what happens when, say, there's a velocity difference of 0.5c between observers and the other sends a photon forwards?
Observer A will measure that in one second, the photon moves about 300,000 kilometres from it's starting point. In other words, typical velocity of light, c.
Observer B will measure the same speed for the photon, but now you need to remember that from A's point of view, B's measure sticks are shorter and seconds take longer. That means that when the observer A sees that B's clock has passed one second, about 300,000,000 of B's one-metre long measure sticks will fit between B and the photon. So A can see that B also measures the speed of light to be constant.
...I've never been especially good at explaining the theory of relativity. It all gets more clear once you derive the Lorenz-contractions a couple of times based on the proven assumption that every observer measures speed of light as constant. There are many ways to explain special relativity, but I prefer first explaining why light will appear to move at constant speed regardless of the speed you're moving at (in relation to other objects and observers) and then explaining a bit what it means and what the interpretations are.
the lights traveling at 1c. if it were a bullet with a muzzel velocity of 1000 m/s and youre traveling at 10000 m/s a second, that bullet will travel at 11000 m/s. but those rules dont apply to light. we like to think in terms of distance per time. but sence time means nothing to light, its always traveling at 1.
Actually, the relativistic addition of velocities can be applied to bullets as well. In such low speeds, however, the result would be something like 10999.99999999998 m/s so it doesn't really pay to calculate the difference... usually. But it's there nevertheless. Obviously, in shooter's reference frame the bullet would be moving at constant 1000 m/s assuming it's a vacuum.
