From loot to gil
The greatest source of money - "gil", as usual in Final Fantasy - is ... other players. To facilitate commerce, Square Enix invented the Auction House, a place for players to list their items and sell them to other players while they do other things - online or offline. Auction houses are located in Bastok Markets, Bastok Mines, South San d'Oria, Port San d'Oria, Windurst Woods and Windurst Walls. They're emblazoned with a gazelle-shaped icon and are always mobbed by virtual capitalists. To use the Auction House, simply target one of the windows and get going.
Before selling any items, select "Bid" from the Auction House menu to see what your items are currently worth. Find the item's category and then the item itself, and look at its price history. Prices of items can fluctuate wildly, so it's usually safest to list the items for the lowest price on the history. The lower your price the faster it will sell, but don't try to sell it for too much less or you'll probably be found out by bargain hunters. Note that many items, such as crystals, can be sold as singles or full stacks, so make sure you're looking at the right entry and set your price. Note that you can only list seven items at a time, so usually it's best to list your most valuable items first.
How do you get your money? After a while, head to one of the Residential Areas in town and enter. Talk to your Moogle and check your Delivery Box to receive your cold, hard cash. Don't worry if you don't know what that crap is yet - you'll be introduced to it soon after you begin your game.