Most DSLRs have an option to use the back button to auto-focus instead of using the half-pressed shutter release button. This is an option which was pointed out to me by Dewi and after trying it, I wouldn't go back to the old way.
If memory serves (which it often doesn't) the back button has to be switched to AF-ON and the shutter release button has to be set to Shutter Only. The best way to configure this is to Google Back Button Auto-focus for your specific camera.
Back button focusing can be useful if you have a bird that keeps returning to the same perch, or - as below - a parent bird returning to feed a chick.
In this case, I focused on the chick and waited for the parent bird to enter the viewfinder then I pressed the shutter release which just took the picture instead of focusing first, then taking the picture.


Back Button Focusing can also be useful for recomposing your picture without mucking around moving your focus points. Just focus using the middle focus point, then compose the picture and press the shutter release.
Holding down the back button and keeping the focus point on the subject will activate continuous focus mode for moving subjects. This can be tricky as focus will be lost if the focus point moves off the subject. There is a setting on the camera to adjust the time delay before the camera tries to refocus but this is meant to cater for objects coming between the subject and the camera.
There is a very good video clip which explains the various focusing modes on a DSLR here.