Basically put, dodging is moving your tank (usually done best with the mouse), in a pattern as to avoid getting shot by another tank.
An example of dodging
How to dodge efficiently
1) Stay alive
2) Get a good score. If you can't avoid the shots, then you can't kill the other tank.
It is almost required to learn to dodge by looking at the radar, since you can see 360 degrees around your tank, instead of the 60 degrees your view can display. You can also dodge more accurately by using the radar, since the shot paths can be adjusted to show where a shot will go before it hits you (see the example from above).
Dodging is recommended over just jumping away from shots, since your tank hangs, uncontrollable in the air for about 4 seconds while jumping. This makes you EXTREMELY vulnerable to any other tanks in the area. If you jump while playing a league match, it almost always means you are going to encounter death. If you have to jump, try to jump onto a building, or behind the tank that is shooting at you, so he/she'll have to turn around.
A simple dodging tactic used by many players is to try quickly turning left or right and then moving in that direction before turning back to face the enemy tank. This is usually a successful way to dodge a bullet and then put the player in an advantageous position to fire a shot at the enemy (since they turned so quickly that the enemy didn't have time to catch up ;) ).
If you are dodging with a mouse, I would recommend making your mouse faster in the control panel settings of your OS. I would also recommend making your mousebox smaller. This allowed the mouse to move the tank from side to side faster, and then you will be able to dodge with greater ease.
Ways to combat dodging
If you are playing on a server with no jumping (like hepcat), then try to fire spreads of three shots towards the tank. This will make it hard/impossible for them to dodge at close range.
If you are playing at a server with jumps (we'll say jumping, rico, 3 shots), fire two of your shots at the tank. Then, when they turn back from dodging your bullets, fire the third shot at them. This usually causes the tank to jump, where you can easily take them out.
I'll leave the rest for you to figure out ;)