A buffer can be imagined as a device that keeps track of input that has already come forth; once a certain condition is met (such as a short time limit or the completion of some discernable pattern), the input is put into action.
Because these are designed to make your life a lot easier at simple actions by fudging the results a little bit, it is also possible to (in many cases, anyway) fudge things around so that the buffer allows you to perform something that would be harder (often nigh impossible!) to do without its helpful interference.
With buffered input, you can pack in a sequence of things in one go. This is great for all sorts of neat tricks, but the most useful purpose is still its original intention: making input manageable.
The primary function of buffered input is to allow one to perform command motions for special moves and super special moves and such. Some games have pretty long buffers, making them very easy to use (in particular? The KOF series for one).
The longest buffer of all is probably that of Art of Fighting 2.
By splitting apart the command, buffered input allows you to more easily input difficult cancels.
Sometimes, you will find that it is to your advantage to do the command input without pressing the button; then when you see an opportunity, you press the button and the command is ready to go counterattack or stuff the opponent. This can be that little extra edge you need sometimes.