Most auctions run for a specified time and end in an event at which the auction manager will end the auction. Depending on how they run things you will either get the full bid or just a specified percentage paid out to you once the auction ends. Pay attention to the infoboards at your auction to figure this out, or ask the local auction manager! The boards will notify the highest bidder and auctionee of who won.
To place your bid, simply pay the board belonging to who/whatever you want to bid on. Make sure your bid is L$50 higher than the previous bid; this is to avoidL$ spamming on the auctions.
To get a board assigned to you, pick a free board and pay it the setup price; It will then tell you that it has successfully assigned to you and you are ready to go.