Open Auction: The most easily recognized type of auction. Most games implement them with turn order for bidding, though a few do not. Power Grid uses this with turn order for new power plants. Indonesia uses them without turn order for mergers. Keyflower uses them for tiles to add to your village, with meeples being the bidding currency, and only one color(of four) being used per tile.
Sealed bid: Typically what is used for sheriff's auctions for property. In games, everyone chooses their bid and presents it concealed, when all are ready, revealed simultaneously. Sometimes combined with the all pay auction, discussed later. O Zoo Le Mio uses this for its tile auctions, and Money uses it to determine order for choosing groupings to obtain. Scoville utilizes the sealed bid in an all pay format for the order of choice of turn order(which has different benefits per phase of the turn).
All pay auction: This is the least popular type of auction in gaming, because it punishes those that lose their bids heavily. This is a specific segment of sealed bid auction, not much applicable outside of gaming. Revolution! uses these to determine who wins each role for the turn. Dune uses a two player version to determine victory in battles.
Fixed price auctions: These can go two ways, either a fully fixed price, or fixed increments. Modern Art uses fully fixed price auctions as one of its types. Vegas Showdown uses fixed increments for tile types.
Reverse auction: This is pretty much another game only thing. No Thanks! gives players penalty points for taking cards, with a sequence being only the lowest number penalty. On their turn, players either take the card turned up(adding it to their negative points and taking all chips on it), or place a chip on it(reducing their positive points.
That's just a glimpse at auction games. There's a lot more great games with auctions, including Metropolys and Ra.
When you play Social Justice, the world loses.