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If you are using a Canon T1i
For one shot focus see page 64 of your (english) user manual.
I use it for static subjects and for focus and recompose - partially depressing the shutter button to aquire focus and then recomposing before continuing to press the shutter button fully to take the photograph.
AI Focus - see page 65 (down the bottom)
I have never used it.
AI servo - see page 65.
I use this for birds in flight and for motor sport.
I track the subject and partially depress the shutter button to aquire focus and then keep it partially depressed, whilst still tracking the subject, so the camera focus will stayed locked on. When the subject looks good in the viewfinder I then full depress the shutter button to take the photograph.
If you are shooting an on-coming subject in continuous shooting mode the auto focus may not be able to keep up.
One-shot is for focus-lock. When you half-press the shutter release, the focus will lock regardless of camera motion (recomposing, for example) or subject motion. For stationary subjects, One-Shot is the usual preference.
AI Servo is for continuous tracking of subjects that are moving nearer or farther away. While you half-press the shutter release, the focus will continually track the subject that's located under the center AF point. This means that you need to keep the center AF point over the subject at all times, which can be challenging with a moving subject (and if it weren't moving, you wouldn't need AI Servo AF). That, in turn, means that your subject will be centered in the photo, so your composition options are extremely limited. If you turn on the auto-AF-point selection, the camera will attempt to recognize if the subject moves from one AF point to another, but this doesn't work well on anything below the pro-level cameras.
AI Focus is a lame attempt to have the camera automatically detect if the subject is in motion or not, and automatically switch between One-Shot and AI Servo. To say that it doesn't work well is an understatement; it almost never works. Its only value is on the "full auto" (green box) mode, where the camera has no clue what kind of subject you're photographing. The pro-level cameras don't even offer AI Focus. Don't use this.
Doug gave a good explanation of each and is pretty much spot on.
A good auto focus technique is to set the mode to AI Servo and switch to back button focusing. This allows you to have the ability to have continuous focus tracking while at the same time letting go of the button you are using to focus so it acts as single shot when you press the shutter button.