Non-Gamer's Guide to This Post
Auction Houses are an in-game way to sell and buy goods. When fighting monsters, sometimes they drop equipment that is good, but your character can't use it. Selling it to a vendor (a non-player character who sells simple goods like food or weapons) is usually a bad idea, because vendors never give you what the item is worth. So you put it on the auction house and hope that someone bids on it.
I just looked up a horde pet on Thottbot and ended up reading more conversations about how to play the neutral auction houses than anything about the pet itself. So I thought I'd try and explain the neutral auction houses to everyone's satisfaction.
Now, every major city has an auction house (Alliance: Ironforge, Stormwind, Darnassus, Exodar; Horde: Thunder Bluff, Undercity, Orgrimmar, Silvermoon). If you can't find the auction house in any of these, ask a guard. That's what guards are for.
You should also have a basic understanding of how to bid and sell on an AH if you've played the game for more than a week. If you don't, go ahead and poke around. It won't hurt. Just be frugal with your money. They say that players walk away broke or rich from the auction house. It is worth it to scout out deals for things you want.
If you win or lose an auction, you will get the item or your bid money (respectively) in the mail. Mailboxes are in front of the inn in every town.
The cities with neutral auction houses are: Gadgetzan in Tanaris (lvl 40-50), Everlook in Winterspring (lvl 53-60), and Booty Bay in Stranglethorn Vale (lvl 30-45).
Booty Bay: I got to Booty Bay by sailing from Ratchet in the Barrens (lvl 10-25). You can also stick to the roads and maybe run through Stranglethorn Vale. I haven't tried running, though, so be careful.
In neutral towns, if you attack another player (PVP), the guards will kill you. They are goblin-controlled areas, and goblins are a neutral race of traders. (They like to get everyone's money.) They do not allow fighting, and the enforcers that I saw were all level 65.
Also, be particularly careful when flagged for pvp on boats. I was killed on a boat and was auto-resurrected at the nearest graveyard, because boats are effectively different areas than on land, being transport from continent to continent. This was bad for me because my character was under level 20 and a bunch of level 20+ mobs were between me and Booty Bay. They attacked me on sight, and I had to hearth home from the graveyard.
Buying/Selling on Neutral AH
I don't know if the three neutral auction houses are connected. All of the alliance AHs and horde AHs are. Neutral auction houses are where alliance and horde may post mutual auctions. In a normal big-city auction house, you do not get auctions from the other faction. But in a neutral AH, both factions may sell and buy.
If a horde player wants a white kitten, which is typically an alliance-only pet, he or she must buy it on a neutral auction house. There is no other way to do it.
This means that an alliance member must post it on the neutral auction house. In a PVE (player-vs-enemy, or Normal) realm, the horde member may create an alliance character to buy the kitten and post it on the auction house, then switch characters and buy it (or have a friend waiting to buy it as soon as it is posted).
On a PVP (player-vs-player) server, you are not allowed to create characters on both factions. You must remain either horde or alliance. Therefore, you need to find an alliance player willing to post a white kitten for you, or you can continually check to see if one is available, though that's the slow way.
Buying something that you can't get for yourself, like an alliance pet, on a neutral AH usually means very inflated prices. Especially if it's something you asked for. I asked for a Bombay from someone, saying I'd pay around a gold for it, and they listed it for a buyout of 99g, obviously wanting people to bid and drive up the price. Which is, in fact, what happened.
I got a Bombay from someone else, but my brother bought the one on the AH for me for a little over 5g. Poor guy. He was very sweet to do it, though.