A "penny auction" (technically, a "bid-fee auction") is an auction where participants pay a fee for every bid that they place.
There are a bunch of variants, but the most common is the "count-up" auction, where each bid raises the price of the auction by a fixed amount (typically a penny, hence the name). A count-up auction is typically supposed to run some fixed length of time, but a bid near the scheduled end will push the end time back a few seconds.
Why do people participate in penny auctions? Products sold at ordinary free-bid auctions tends to fetch a "fair" price -- that is, the buyer ends up paying approximately what he would at a fixed-price retailer. It's difficult to get a much better price, simply because if the price were lower, other people would jump in and bid it up.
In a bid-fee auction, other bidders are reluctant to "jump in" -- they would have to pay to do it. As a result, 80% and 90% discounts are not uncommon.
The paradoxical effect though, is that penny auctions are popular because they are unpopular. People participate in them because they think other people won't! And somehow, it seems to work.
Look at a recent example. BidCactus, a popular penny-auction site sold a Sony Blu-Ray Disc Player that listed at $299.99 for $67.20, more than three-quarters off. For this auction, bids are $0.75 and the price jumps up by 10 cents with each bid. The BidCactus people must be happy, they were paid $571.20 (that is, the $67.20 final price plus the 672 bids at 75 cents each) for a $300 video player. The winner, who calls himself "burritoman", must be happy, he got a huge deal. The other people, the ones who didn't win (let's not use the L-word here), well, let's hope they had fun.