Say that Jonathan Papelbon always throws his fastball at 100 miles per hour. Suppose that someone puts him on a skateboard, and starts the skateboard moving towards you at 15 miles per hour. Papelbon then pitches a fast ball straight at you. If you measure its speed, you will of course find that it's going at 115 miles per hour.
Now suppose that instead of Papelbon, we put a laser on the skateboard. Does the beam of light behave the same way? In other words, would we measure a faster speed for the light when the skateboard is moving towards us than when it's sitting still?
The answer turns out to be No: the speed of light is the same whether the source of light is moving or sitting still.
You might think this would be difficult to measure. After all, the speed of light is extremely large, so any change in it that you'd get by moving the source around would probably be quite small in comparison. But using astronomical observations, it turns out to be quite easy to test this. The key is to use observations of binary stars.
Suppose we observe a binary star system, which is just a pair of stars orbiting around each other. For simplicity, let's suppose that we focus our attention on just one star in the pair. (Maybe one star shines much more brightly than the other, for instance.) This animation shows how the light from that binary star travels from the star to us.
In this picture, the solid white circle a the right is the star. The dotted circle is the star's orbit. The vertical white line at the left shows our own location. The red represents light traveling from the star towards us. (At each instant, the star emits light in all directions, but all that's shown in this picture is the light that happens to be traveling in our direction.)
If you like, you can imagine that the star is a pulsar, which is a star that emits bright pulses of light (actually, radio waves) at regular intervals. In that case, the red dots represent the individual pulses. Or you can just imagine a regular star that emits a steady stream of light. In that case, the red dots don't signify anything in particular; the continuous red line is what matters.
In this animation, I've assumed that the speed of light is constant. In other words, all the red dots travel towards us at the same speed. (This is the way the world really is.)
If we stand at the left of the picture and look at the star, what do we see? At any given instant, the star appears to us to be located at the place where the light (the red line) hits our location. That is, the apparent position of the star at any given time is represented by the location of the green asterisk. Note that we see the position of the star wiggle up and down pretty much as we expect.
There is one small but interesting effect. The oscillation of the star is not a pure sine wave: it wiggles more quickly at one end than at the other. That's because the star's orbit takes it closer to and farther from us. That means that the light from the star gets to us a bit early in some parts of its orbit and a bit later in others. That effect is real, but it's an incredibly tiny effect for actual star systems. In this animation, I deliberately made the speed of light pretty slow and the speed of the star in its orbit pretty fast, which exaggerates this effect.
Now let's examine what would happen if the speed of light depended on the motion of the source (that is, if light were like fast balls). In that case, the light emitted when the star was at the top of its orbit would travel towards us faster than the light emitted when the star was at the bottom of its orbit. The result would be something like this:
Things would look very strange to an observer in this case. Most of the time, she would see multiple images of the star. The reason is that some of the light travels so fast that it catches up to and passes light that was emitted later. At any given time, the observer sees light from multiple different orbits. In this particular case, the observer usually sees either three or five images of the star.
As I mentioned before, this animation is made with a very slow speed of light, so you might wonder whether this effect would be significant for real star systems. The answer is that it turns out to be extremely significant. The reason is that the stars are very far away. That means that, even though the difference in the light's speed at the top and bottom of the orbit would be very slight, the faster light would still have lots of time to overtake the slower light. In fact, in real star systems, you'd end up seeing not just three or five images of the star, but thousands of images.
Needless to say, we don't see that sort of thing at all, and that provides extremely strong evidence that the speed of light does not depend on whether the source is moving towards you or away from you.