The word proof has a lot of meanings. It can refer to alcohol content, a mathematical procedure, or evidence. However, in the world of baking, proof refers to rising dough.

Home bakers, and recipes aimed at home bakers, refer to it as "rising," but professionals call it "proofing." Proofing is a part of the bread making process where fermentation takes place, causing the dough to expand, grow, or rise (whichever term you prefer). Depending on who you ask, proofing can include the first period of fermentation, which is usually called the bulk fermentation. Most of the time, though, proofing is the second stage of fermentation, called the "second rise" by home bakers. This is the period after the final shaping when the dough is left to expand to it's proper size before baking.

Proofing is an important stage in baking (of course each stage is important in its own way) for three reasons. First off, more fermentation occurs for that much more flavor. The fermentation then causes gas production which makes the dough expand to a larger size. Also the gluten is able to relax which allows oven spring to take place.

Try as I might, I could not find the origin for proofing in reference to baking. If I factor in the fact that some bakers call it "proving" and all the definitions, though, I think I can hazard a guess. Though I have no idea why or where it started, I think this stage must be called proofing because the expanding dough proves that the yeast is alive and working. That makes sense if you think that until modern times all bread was made with wild yeast. Bread making was a much more chancy activity because they couldn't be sure that they had yeast activity, unless they had a starter culture that they were sure of.