Once the giveaway ends, the site will randomly pick one person out of all the people who entered (currently 5659) to win. Then, the giveaway creator will give the key, link, etc. to the winner and the giveaway will be concluded.

As long as you click the enter button, you will be counted as an entry and you have a shot at winning the game regardless of whether or not you are online at the time that the giveaway ends.

Hope that clears some things up.

To follow on from what Decimation008 said, if you enter for the game but do not get picked, nothing will happen and you can continue entering for other games as per usual. However if you are the winning draw, you will get a pop-up message (and an alert in the bar at the top), and once the giveaway creator has sent you the stuff (can be delayed depending on the method of giving you the key), you go there to mark that you have successfully received the game.

The points you use to enter giveaways regenerate over time.
Also, you are required to redeem the game to the account you have linked to your profile here. Selling it, trading it, giving it to someone you know, or just outright not marking as received once you have claimed it will get all kinds of wrath directed at you.

It's a pretty simple system really!
When in doubt, refer to the FAQ : https://www.steamgifts.com/about/faq

they are just thanking for the giveaway, nobody has won yet