If I have a snipe set on an auction, do I have to leave my computer on with JBidwatcher running for this to work?

Yes. In fact, if you try to exit JBidwatcher when you have set a snipe to happen, it'll warn you about that.

The JBidwatcher software places the bid directly with the eBay site, when it's time; in order for that to work, JBidwatcher needs to be running, and the computer needs to be on, and awake. Computers that automatically go to sleep in such a way as to shut down networking, and/or not let software continue running, will interfere with JBidwatcher's operation.