It's one of the first technical lessons any budding photographer learns: Focus. Recompose.
And it's a bunch of crap.*
In case you missed that day in class, focus-recomposing is a method of focusing on a specific subject by pointing your camera specifically to what you want to focus on (usually using the center focus point), and then while holding that focus, moving your camera to frame your shot. Most cameras by default have you press your shutter half-way down to activate focus and then the camera guesses on what is important. If you've got a little point-and-shoot camera with face detection and you only ever use it for taking pics of your friends at the club, then this article is not for you. That method will work just fine. But, if you need any control over your compositions, then listen closely. First, a diagram explaining why this method doesn't work:
A picture is worth a thousand words, but I should clarify that that peach circle should represent a person, and optimally we would be focusing on their eye that is closest to the camera. In this case, just think about focusing on point of the circle closest to the camera. The red line represents the plane of focus, meaning everything in red would be perfectly focused, and everything not red would be blurring out.
Before going further, I will say *I'm indeed being overly hard on this technique. In many shooting situations (such as using wide angle lenses, or just about any lens stopped down to f/5.6 or f/8) there will be no discernible difference in the final result. The major issues lie in using large aperture lenses (f/1.2, f.1.4, f/1.8) on a close subject. In these situations you are dealing with razor-thin focus planes and even the slightest shift is the difference between an eye being in focus, or an ear being in focus. This is the difference between a composition that works, and one that doesn't.
So focus-recompose works sometimes. Sometimes it doesn't. But because of this, it's generally a bad habit. So what's a good habit? A good habit is using manual focus points and selecting the best one without altering your framing. It takes a little longer, but accuracy is better than speed. Besides, once it becomes second-nature, it'll be fast. You'll begin to preset the focus point to the side of the frame you will need before even bringing the camera to your eye. The bottom line is that you want your camera to move as little as possible once you've selected your focus.
Bonus tip: By default, nearly all cameras attach the auto-focus function to the shutter. You know, press the shutter down half-way and it focuses. Did you know you can detach this function and move it to a dedicated button on the back of the camera (usually a button labeled AF-ON or AF-L)? Sometimes this technique is called "back button focusing," but regardless of its name, it's a huge boost in control. There are many times you want to be able to set your focus and then not worry about losing it because you let go of the shutter. One of the most popular needs of this function is if you do a lot of panoramas... but it helps out in everyday shooting more than you think. Once you get used to this method, the old way seems downright clumsy.