What you see below (that is, if your browser is Java-enabled) is a simulation of a simple physical experiment. An electric charge moves at the bottom of the window. It produces an electric field that is measured by the observer at the top of the window. The red arrow shows the direction and the intensity of the field at the observation point.
You have to imagine that you are either viewing the experiment in very slow motion--so slow that you can see light propagating from one point to another (the expanding circle is the wavefront of light)--or performed on cosmic distances.
Since information cannot propagate faster than the speed of light, the observer has no way of learning of the "actual" position of the particle at the moment of the measurement. It can only know about the last known position of the particle--the position it had at the time when it emitted the light that has only now reached the observer. That position, also known as the retarded position, is depicted by a hollow circle. Notice how it always trails the actual position of the particle.
Naively thinking, one would expect that the electric field at the observer's location should point towards that trailing image. That's where the observer actually sees the particle.
Amazingly enough, most of the time the field points towards the estimated position of the particle, which is pretty close to the actual position.
When the particle moves with a constant speed--as it does here between the two notches on the trajectory--it is easy to estimate its position, and you can see the arrow (and the projecting ray) doing a pretty good job around the center of the trajectory. But we made it more difficult--when crossing the notches the particle suddenly starts decelarating. The observer has no way of learning about it in "real" time (if there was a notion of "real time" in special relativity). To emphasize this, we draw the wave of light propagating from the turning point towards the observer. Only when this wave hits the observer, can he possibly detect the change in the movement of the particle. And, as if by magic, the field suddenly changes its direction as if to reflect the deceleration that's been going on for some time.
Continuing deceleration finally makes the particle stop and then turn back. The field measured by the observer pretty well reflects that. But at some point the particle hits the notch again and the acceleration suddenly stops. Again, the observer has no way of knowing that, and the field slowly gets ahead of the particle. Only when the wave emitted at the turning point hits the observer, there is a sudden adjustment.
For the purpose of demonstration, the accelerating particle changes color from blue to red. The color of the observer reflects not the current acceleration state of the particle, but its observed state. So the observer turns red only after the information about the acceleration has arrived on the back of a light wave.
This is a really weird and unexpected result. It's not like the observer is doing some calculations. It's the electromagnetic wave, which is hitting him, carries the adjusted-for-speed-and-acceleration information. Why would the direction of the electric field follow the estimated rather than the retarded position? Imagine for a while that the opposite were true and the field of a uniformly moving particle laged behind it.
Now think of a very large particle (a planet or a galaxy), in which the front may interact with the back. If the particle is uniformly charged, the back of the particle is repulsed by the front and vice versa (it doesn't split apart only because gravity holds it together). But when the particle moves, the field generated by the front would lag behind, so the back of the particle would fill it closer (and stronger) than it really is. The front of the particle, on the other hand, would see the back lagging farther than it really is. The result would be that the front is repulsed less than the back--there would be a net force trying to slow the particle down. That can't be right! Charged particles don't slow down!
So here's what Nature does through Maxwell's equations of electromagnetism: it adds a second term which anticipates the movements of the particle. So if the particle moves uniformly, this term will adjust for it.
And then there is a third term, which takes into account the acceleration. This term causes the jerky movements of the field.
For the mathematically inclinded, here's the full formula:
Download the Java source for the applet.