For the purposes of this research, we will consider two basic auction types: first-price and second-price.
In a first-price auction, the bidder with the highest bid wins the item, and pays the price she bid. This category of auctions includes a number of different auction techniques. It is sometimes referred to as a "Dutch Auction" because flower auctions in the Netherlands are conducted by announcing first a very high price and then announcing gradually decreasing prices. The winning bidder calls out when the price has decreased to an acceptable point and no one else has yet made a bid. Another common type of first-price auction is the sealed-bid auction commonly used to award contracts. Auctions in which the bidder is paid a price, and the lowest bidder wins, are generally treated as being mathematically equivalent to highest-bid auctions.
A second-price auction is one in which the bidder with the highest bid wins the item, but pays the price offered by the second-highest bidder. This may seem a strange way of doing business at first, but it is basically equivalent to an ascending-price auction such as the ones conducted for artwork and used cars. Since in such an auction, all bidders except the winner drop out when the second-highest bidder's price has been exceeded, the winner ends up essentially paying the second-highest price offered, plus some relatively small amount. A sealed-bid second-price auction is referred to as a Vickrey auction, after its inventor.