It is a piece of gear (normally a rack mount unit) with several female jacks - which allows you to route signals for one piece of gear to another.
As an example - let's say you have a couple of effects modules which you want to use to add delay to a keyboard part or vocals, etc. Rather than crawling behind the gear to plug the cables in and out - you always leave the effects units plugged into the patch bay - then you simply plug (patch) the keyboard signal into which patch point you want.
Patch bays come in different sizes to accomodate 12 or 16 or 24 pieces of gear (or even more). Bays also come in 1/4" (the most likely thing found in semi-pro studios) and also can be found with RCA jacks, XLR jacks and other configurations.
If you've even seen an old movie where the phone operator connects calls by pluging cables into a big board - thats a patch bay.
