Auction - you have an imaginary budget to build either your entire team or build your starting roster and the bench is a draft (leagues vary). Most leagues do either 100 or 260 but you can do whatever you want. Auctions IMO are much better than drafts because you have full control on how to build your team and avoid a lot of luck of the draft. You cant draft say Kemp and Miguel Cabrera right now, today, but you could get them both in Auction. In a set order, each owner throws a player and a starting bid. Either your host site software or your auctioneer will track the goings on and at the end of the bidding if you have high bid you get him.
Owner A- Matt Kemp $10
Owner B - $30
Owner C - $40
Owner A $45
Owner B - $46
Auctioneer : Going Once, Going Twice
Owner F: $50
Auctioneer: Going Once, Going Twice...sold to Owner F for $50.00
Owner B: Papelbon 1$
Owners are required to leave at least 1 dollar for every spot they have to fill (you can easily make some spreadhseets etc that calculate max bids or again your host site will track that if online)
FAAB - every day/night all free agents are locked and in order to get the new closer of a team, or guy who hit 3 HR last night everyone has an imaginary budget of a varying amount (many times$100) to bid on said new player as opposed to the waiver order or first come first served. There are two different ways of bidding. "ebay bidding" where if you bid say $50 and the other guy bids $75..he gets him for 51 or the other way he would get him for $75. You can also vary who exactly goes into waivers. some leagues do it for cut players only, some leagues only have it once games are started, some leagues have it constantly.
Dynasty - yes, basically a bigger keeper format. You keep more players usually vary the pricing/contracts more, usually have more minor league rosters etc.
Don't want to generalize things but in the order I have them listed is how applicable they are for casual to hardcore fans. As in for casual folks FAAB and dynasty formats can be way over their heads (so isnt auction to a lot of casual FB players)
"I'm the man with the ball. I'm the man who can throw it faster than F***. So that's why I'm better than anyone in the world." - Kenny Powers