http://steamcommunity.com/tradingcards/
If you own a game that has cards for it, you launch the game and while the game is launched you will randomly get card drops. You no NOT have to be playing the game. I launch most games and alt tab them and let them run in the background doing my normal stuff and you will still get the card drops. Each game has x amount of cards that will drop for you but will require xx amount to be able to craft a badge for that game. Example. Bioshock Infinite has 7 total different cards, you can only get 4 as drops. So if you do not receive any doubles/+ you will still need to get the 3 other cards you don't have to be able to craft the badge. You can get these cards either by trading with people or buying them off the Steam Market, alternativly you can sell the cards (or just your doubles/+) on the market as well to get Steam Wallet money when someone buys them from you. There are foil versions of cards which are the same as normal cards but a lot rarer and you will just want to sell those on the market because they will fetch a lot more money than a normal card and you can only craft foil cards with other foil cards.
So you have all the cards needed to craft a badge. You hit craft, it does some mumbojumbo magic fuey and boom, you get 100xp towards your Steam profile, a badge of the game you crafted it for, a background and emoticon of the game you just crafted it for as well. During this Summer Sale event you also receive a Summer Getaway card that when you can only get during this sale, it has its own badge and requires 10 different cards (you can get these Summer Getaway cards by 2 other methods as well, first being for every $10 you spend in the Steam store during the sale you get one, and for every 3rd community vote you vote on you will get one). Normally instead of the Summer Getaway card you would receive a coupon for a random game at a random % off.
Hopefully that clears things up a little bit.