if you mean types, then in terms of auctioneers and auction items, we can differentiate three types of auctions:
exchange auction - also known as commodity auctions or exchange-commodity auctions, are the most closed to the new participants. The participants include a number of core professional buyers, who monitor each other to ensure that no one is 'cheating' on the community
sale auction - for art and one-of-a-kind items
dealer auction - for collectibles
A more detailed differentiation would be between:
English auction: This is what most people think of as an auction. Participants bid openly against one another, with each bid being higher than the previous bid. The auction ends when no participant is willing to bid further, or when a pre-determined "buy-out" price is reached, at which point the highest bidder pays the price. The seller may set a 'reserve' price and if the auctioneer fails to raise a bid higher than this reserve the sale may not go ahead.
Dutch auction: In the traditional Dutch auction the auctioneer begins with a high asking price which is lowered until some participant is willing to accept the auctioneer's price, or a predetermined minimum price is reached. That participant pays the last announced price. This type of auction is convenient when it is important to auction goods quickly, since a sale never requires more than one bid. The Dutch auction is named for its best known example, the Dutch tulip auctions; in the Netherlands this type of auction is actually known as a "Chinese auction". "Dutch auction" is also sometimes used to describe online auctions where several identical goods are sold simultaneously to an equal number of high bidders. Economists call the latter auction a multi-unit English ascending auction.
Sealed first-price auction: Also known as Sealed High-Bid Auction or First-Price Sealed-Bid Auction (FPSB). In this type of auction all bidders simultaneously submit bids so that no bidder knows the bid of any other participant. The highest bidder pays the price they submitted.
Sealed second-price auction, also known as a Vickrey auction: This is identical to the sealed first-price auction, except the winning bidder pays the second highest bid rather than their own. In theory, this is mathematically equivalent to the English auction, because in both the first-place bidder receives the item at a price equal to the second-place bidder's willingness to pay, plus the bid increment. True strategic equivalence requires a modified model of the English ascending auction in which the price rises continuously with bidders choosing when to drop out. When all but one bidder drops out, the good is allocated to the remaining bidder at the price at which the second-to-last bidder dropped out. Implemented as such, this is known as a Japanese Auction.
Silent auction: This is a sealed variant often used in charity events, but involving the simultaneous sale of multiple items. Participants submit bids normally on paper, near the item. They may or may not know how many other people are bidding or what their bids are. The highest bidder pays the price they submitted.
Procurement auction: This kind of auction reverses the roles of seller and buyer. The buyer puts out an RFQ for a given commodity and providers offer progressively lower prices in hopes of getting the business. At the end of the auction, the lowest bid wins.
Digital art auction: In this indefinitely long auction, designed for unreleased works that are trivially reproducible at zero cost (recordings, software, drug formulae), bidders openly submit their maximum bids (which may be adjusted or withdrawn at any time). The seller may review the bids and close with a price of their choosing at any time—the successful bidders that pay this price are those whose bid meets or exceeds it, and these are the only bidders who receive a copy of the item.
Open outcry auction: This type of auction is used in stock exchanges and commodity exchanges, where trading occurs on a trading floor and traders may enter verbal bids and offers simultaneously. Transactions may take place simultaneously at different places in the trading pit or ring. This type of auction is being replaced by electronic trading platforms.
Unique bid auction: In this type of auction users post blind bids and are given a range of prices they can place a bid in, often a capped limit. The highest, or lowest, unique bid wins. For instance an auction is given a maximum bid of 10. If the top five bids are 10, 10, 9, 8, 8 then 9 would be the winner being the highest unique bid. This a popular online type of auction.
Buy-out auction: This auction has a predetermined buy-out price in which the bidder can end the auction by accepting the buy-out price. The buy-out price is set by the seller. The bidder can choose to bid or use the buy-out option. If no bidder chooses to utilize the buy-out option, the auction ends with the highest bidder winning the auction.
Combinatorial Auction: In some cases, a buyer's value for the goods that are up for auction is a complex interaction of the type and number of goods he receives (known as a "bundle"). For instance, if bicycle wheels and bicycle frames are sold separately in an auction, a bidder may value a bundle consisting of a single wheel or a single frame at $0, but may value the bundle of two wheels and one frame at $200. If forced to purchase each component of a bundle in a separate auction, the bidder faces a dilemma: bidding enough to win the components of the bundle that are sold first may result in a financial loss if he fails to win the components that are sold later, but failing to purchase the components that are sold first ensures that he will not win the bundle. This dilemma can be overcome by selling all goods simultaneously and allowing buyers to submit bids on combinations of goods. Such combinatorial bids may offer to pay a certain amount if all units of a buyer-specified bundle are awarded, but nothing otherwise. They may also offer to purchase one bundle of goods or another, but not both. Sorting out which buyers win which bundles (and sometimes the amount they must pay for them) is usually computationally complex. This complexity is overcome by feeding the bids into an optimization algorithm (such as a linear programming problem).
If more than one identical item is sold, there are two possible generalizations of the second-price auction. In a uniform-price auction, all of the winning bidders pay the price submitted by the highest non-winning bidder. Bidders will not typically bid their true value in a uniform-price auction with multiple units. In a Vickrey auction, the pricing rule is more complicated, but preserves the property that bidders will bid their true valuation. It is also possible to auction each identical item individually. Once each item has been priced, the winning bidder is entitled to buy the remaining goods at the same price. Items the winning bidder opts not to purchase are auctioned again. This system creates a tension between the desire to hold back on bidding since later items will almost certainly be cheaper, and the chance that by losing the first round of bidding all possibility of purchasing will be lost.
Bidders in the traditional Dutch auction and sealed first-price auction will tend to underbid what they believe the item is truly worth in hopes of getting the item for less, or in order to avoid the winner's curse. This behavior is known as bid shading. These two auctions are also theoretically equivalent, but in practice Dutch auctions will produce less revenue than sealed first-price auctions (one of the important results of Experimental economics).