Burst mode, also known as continuous shooting mode, is where you take multiple photos in quick succession with the same settings. You setup as if you were going to take a single photo, but by being in burst mode and holding down the shutter button, you can take multiple shots.
Most consumer digital cameras and all digital SLR cameras released in the past few years have at least some burst capability. The newer and nicer your camera, the more likely it's going to be able to do a faster burst and for longer. Look for a symbol of rectangles overlapping each other in the user manual or in menu of the camera.
When you shoot sport photography in burst mode, you have more frames to choose from. You can capture a sequence of pictures that are frames apart and this allows you to then analyze the sports motion. It is ideal for studying a golf swing or pitching motion. You want to make sure you have plenty of memory in your compact flash card as you will need it when you start shooting three to five pictures at once. When photographing fast moving subjects you will need to make sure you test it a few times to get your timing down. You will not be able to capture the entire motion but you may be surprised at some of the images you get, so give it try.