The "weight" of a pass is how hard you hit the ball. As far as English language is concerned, the meaning is because of the association between weight and force.
Weight of pass is key:
The speed of the ball is crucial to a successful pass
A "well-weighted pass" is one that has the correct weight for the situation. It's usually used when the pass must be neither too fast nor too slow in order to work. Of course, with Xavi's standard and style of play this is all passes ;-) A pass hit forward as hard as possible is not usually referred to as "well-weighted" even when it's correct to hit it that hard.
Note that a "well-weighted pass" is a pass that is "well-weighted", not a "weighted pass" that is well :-)
It is most important to weight the pass correctly when passing into space, hence the meaning given by J.R, that a "weighted pass" is a pass where the ball meets a team-mate running onto it. That is to say, a pass where the weight is just as important as the direction and flight. Too hard and your team-mate won't reach it. Too slow and there may be a defender in position to cut it off, or even the best outcome your team-mate must break stride to collect it.