DSLR cameras, by default, set focus by pressing the shutter button halfway down. Back button focus simply changes the method of focusing by assigning the focus function to another button on the back of your camera (this button will differ depending on your camera model). This means that your index finger is now solely responsible for releasing the shutter, and your thumb is now responsible for focus.
When you change the focus button, you free up your shutter button to only be responsible for releasing the shutter. This may not seem like much of a game changer, but think for a moment about photographing a moving subject (sports or family and child photographers, anyone?). In the millisecond it can take to set focus and then release your shutter, your subject may have moved. The result: a blurry and out of focus subject. By separating the focus and shutter functions, you can set focus and get the shot simultaneously. Gaining back those precious milliseconds via back button focusing allows you to nail the focus.
Back button focus alone is beneficial, but if you frequently photograph moving subjects, you can really take your ability to nail focus up a notch by utilizing a continuous focus mode along with back button focus. A continuous focus mode (AI Servo for Canon, AF-C for Nikon) allows you to track a moving subject and keep it in focus. By continuing to press the focus button, your camera will automatically readjust focus as your subject moves. In the photo below, I was able to track the movement of my daughter running towards me by keeping my thumb on the focus button as I released the shutter with my index finger. Even though she was running towards me and the plane of focus was changing by the millisecond, by utilizing back button focus and the continuous focus mode together, I was able to achieve excellent focus on an otherwise difficult shot.