Just head to our giveaway page and pick up your key. Register on the Eligium home page if you haven't done so yet, and then you can go to the redemption page to enter your key. That's all there is to it; you're set for download and launch!
You'll have plenty of time to play and content to experience, as the closed beta lasts until January 18th and has a level cap of 45. Make sure you take some time while you're installing to check out the closed beta forum, and have fun in Eligium!