The phenomenon of aberration can be explored by using simple examples. A common scenario is the train/car (I'll stick with the car) and rain. If you are riding a car, you notice that the raindrops appear to be coming from ahead of you and heading towards you (equivalently, if you look through your windows at the side, the rain appears to be inclide at a certain angle). As you travel faster, the angle of inclination increase. To someone standing still, however, the rain is simply falling down vertically. This is aberration.
We can also observe this effect in photons, or in light, and hence in light sources such as stars. In this respect, we refer to it as aberration of light. Aberration of light happens in the same way, in fact, it's happening as we are moving. The aberration only depends on the speed of light and the speed at which you are travelling. The speed of light is very fast! Classical aberration will predict that the observed aberration is in the order of v/c where v is your velocity, and c is the speed of light. This value is almost negligible in everyday life that we don't usually observe it.
However, as you increase your speed and approach the speed of light, the effect becomes apparent, but classical physics now fails to account for the aberration. Near the speed of light, relativistic effects take effect, and we now have relativistic aberration of light.
Now, let's imagine we're riding a spaceship (a very futuristic one, able to travel near the speed of light). At rest, we see the normal stuff we see, no changes. As we start to move, we see that they become slightly distorted, like the sides are going towards the center (the effects are negligible, of course, as long as we are not in high speed). This continues until we reach about half the speed of light! At this speed, objects that are `120^o` from our direction are shifted, and we see them towards our front. At velocities very, very close to the speed of light, everything appears to be approaching towards, except for the things that are directly behind us! (you've probably seen some of this when watching sci-fi movies, the light from the stars appear to be clustered in front of the speed's main dock).
The effect of this aberration of light is that since the earth is moving, the location of the stars appear to be shifted by certain angles from their actual position.
If you want a visualization of my explanation on relativistic aberration, nice illustrations are provided in my first reference.