The Diablo 3 Auction House is probably the most controversial thing about Diablo 3 as of date. Why? Because it will allow players to sell items for real money.

That's right, the Diablo 3 Auction House will allow players to sell a variety of things in-game for real money. Now before you think this is just a way for Blizzard to make more money from the game, read the full article.
Here are some of the features that will be in the RMT (Real Money Transfer) Diablo 3 Auction House:
- Allows players to sell their gold
- Allows players to sell their characters, althou this won't be in the initial release
- Can be accessed in game just like a regular auction house
- The auction house will be 100% player driven, Blizzard won't be selling their own items on it
- Differenet Auction House for different currencies such as Euro, Dollars, etc
- Similar to Ebay but for Diablo
- Auto-Bidding for when you don't want to loose an item
- Anonymous bidding and selling

Some more things you need to know is that there will be fees for listing and selling items, but this fee will be fixed. So the fees for selling a $200 item will be the same as if you sold a $10 item.
However, selling a few items per week does not require any Real Money investment from the player, as each player is given a weekly "Free Listing" allowance.
The money you get from selling items is stored in your account, which can then be transferred to either a third party tool or to your Battle.Net account. Transferring to your Battle.Net account will allow you to buy items from the Battle.Net store, while transferring to a third party tool will allow you put the money on your credit card. The third party tool that will be used has not been announced yet but it will be something similar to Paypal (if not Paypal).

Also, the Diablo 3 Auction House (the RMT version) will not be usable by Hardcore players, but they can still use the gold version. The reason Blizzard gave for this is that they don't want people spending money on a character than will be lost when they die (remember that Hardcore mode has permanent death).
Now as to why Blizzard actually decided to do this, is twofold. First, it allows Blizzard to control RMT transactions and second it allows Blizzard to make more money. By controlling RMT transactions I mean that instead of players buying gold or items from shady third party sites like they did with Diablo 2 and many other games, they can now buy them safely in-game.
Now this is all fine and dandy for people who have a tendency to buy gold with real money, but what about people who would not have done this anyway? Are they now forced to pay real money for items?
The answer is no, as there will be both a RMT auction house and a gold auction house.
Here is the official reason Blizzard gave for implementing this feature:
We’ve always tailored our business models to match what we’ve felt would be most appropriate and effective for each game and in each region, and that’s the case with Diablo III as well. The item-based nature of Diablo gameplay has always lent itself to an active trade-based ecosystem, and a significant part of this trade has been conducted through unsecure third-party organizations. This has led to numerous customer-service and game-experience issues that we’ve needed to account for. Our primary goal with the Diablo III auction house system is for it to serve as the foundation for a player-driven economy that’s safe, fun, and accessible for everyone.
Of course, there is a major downside to this. With this new implementation, your payment information will be stored in your game account along with all the other details. This means that if you get hacked, you are in real trouble. You can decrease the chances of this happening by following proper safety procedures and using a Blizzard authenticator.