The Auction House is run by Lloid, a talking Gyroid similar
to the ones from Animal Crossing that served as your mailbox
and auctioneer. Here, you are able to put items up for bid, and
potentially get more bells for them than you could from Nooks or
by selling online. This can work in your favor especially well for
items that can't be bought, like fossils.
The Auction House runs passive auctions. That is to say, you can't
see what other people have bid on an item, and can place only one
bid on it yourself. As you're not actively competing with the other
bidders, your best bet is to bid as high as you're able.
Bidding periods within the Auction House begin on Saturday at
6:00 a.m., and end either on Sunday at midnight, or Monday at
6:00 a.m. (confirmation needed).