Imagine you've struck your tee shot and also hit the approach to the green, but your approach shot comes up just short of the putting surface. If you make an up-and-down, however, you can still make par. What you need to do is get the ball up onto the green with one stroke, and then down into the cup with another. Up and down.
Technically, you can use "up and down" to describe any two strokes that result in the ball going into the hole. But typically, "up and down" is almost exclusively applied to shots from just off the green and from greenside bunkers, situations where using only two strokes to hole out is the most likely positive outcome.
He sank the putt to get his up-and-down.
Tiger Woods needs to make an up-and-down to save par.