Battle.net Balance is the currency that will be used to make purchases through Battle.net with real money. This includes the real money Auction House but also games, items and subscriptions from the Blizzard Store for other Blizzard products.
Money is added to the Balance primarily with a debit or credit card after which it can be used. Money once converted cannot be transferred back as cash to the player, but must be used to purchase products with. The only way to convert Balance back into cash is through the Auction House; by turning the Balance into items which are then sold on the Auction House the playe can receive money. This transaction scheme will however incur several costs along the way, listing fees, selling fees and the conversion rate employed by PayPal o similar services, which will make it a problematic way to convert Balance into money.
Players do not have to use the Battle.net Balance for any form of purchase howver; it is not even necessary for the Auction House. It will however be much easier to use through the interface.
Battle.net Balances of different currencies are tracked separately, thus one unit of Balance bought with USD is not equivalent to one Balance unit bought in EUR or any other currency.
There is an, as of yet, unspecified limit to how much Balance one account can hold. Some regional laws may require Blizzard to empty the Balance on any accounts that haven't been accessed in over three years. At this time it is unknown which regions this concerns.