The first three, or big three as I like to call them, are all complete auction mods that handle many things related to the auction and one handles a whole lot more than just auctions. You can use just one or use them all, they play well together. Most players probably won't need or want all three though there are those who do use them all regularly, preferring one for posting, another for buying and the other for something else. If I had to choose just one of the big three to recommend it'd be TSM.
Tradeskillmaster (aka TSM) - More than just an auction mod TSM is a very complete modular integrated package that does everything you used to need a whole bunch of different addons and scripts to do. It has optional modules to handle auctioning, crafting, mailing, accounting, database, shopping, gathering and more. I'd say TSM is preferable for big auction players listing thousands of items daily and certainly for undercutting wars.
Auctionator - Another auction mod with it's own unique and effective way of doing things. Good for both buying and selling auctions. You might prefer Auctionator over TSM for buying or selling auctions. In particular a lot of players seem to prefer Auctionator for buying auctions over anything else. Auctionator's selling interface is pretty good too. I'd say Auctionator might better suit the player who isn't posting a real lot of items every day, likes to pick his prices more carefully and doesn't always undercut. A great choice for auction players who aren't ready for the complexity of TSM yet and need something simpler and more intuitive.
Auctioneer - The grandaddy of auction mods. Still pretty good for buying, sorting auction lists and has good database functionality that can be accessed by many other addons too. It's long been surpassed by other addons when it comes to posting auctions. Unfortunately Auctioneer is getting rather long in the tooth now. I can't really recommend it for anything other than the snatch list, database, sort and search functions. In fact Auctioneer's "protect auction window" might be it's single best feature now (it's the main reason I haven't uninstalled yet) but it'd be silly to recommend installing such a large mod just for that.
While not auction addons per say the following small addons improve your organization, speed and efficiency by making repetitive chores much simpler.
Bankstack - One click bag and bank sorting and stack compression.
Postal - Very configurable mail opening addon with a plethora of useful features such as automatically opening different types of mail and refreshing the mailbox to get more while you go make coffee or something.
Dumpster - Great for quickly moving items to and from the bank. Almost a must have for glyph posters!
Addon Control Panel - This one won't do much for your auctioning game but it's great for quickly turning different mods on and off in game without logging out.