As far as Canons go, there are three focusing modes, One Shot, AI Focus, and AI Servo. In my opinion, AI Focus is practically useless.
One Shot is just what it sounds like. When the shutter button is depressed halfway, focus is attempted, and when achieved, it is locked. You can then swing the camera all around and focus will not change.
AI Servo continuously focuses while the shutter button is depressed halfway, and you will hear and/or feel your lens' focus system going nuts while you're holding the shutter button halfway down and moving the camera around. The concept behind AI Servo is to keep focus at all times, though your mileage may vary based on your lens' focusing speed, camera's AF capability, and the circumstances of your scene.
AI Focus just does One Shot until the AF system senses that the subject has moved (or you're swinging your camera around), at which time it switches to AI Servo. If you're in a situation where your subject may be moving, just choose AI Servo.
It's definitely not a silver bullet, but it's probably the best bet for most situations, simply because manual focus is very difficult to achieve quickly and AF systems are pretty darn good these days. I have used AI Servo to track animals and people and have had good luck with it.
As for burst (or continuous drive) mode, yes, AI Servo focuses continuously regardless of the drive mode as long as the shutter button is halfway or fully depressed.