I'm a Hardened or Prestige Edition owner, how do I get Moon for Free?

If you have already downloaded the Zombies Pack that came with the Hardened and Prestige Editions of Call of Duty®: Black Ops, fire up the game and go to the Zombies menu. Select either SPLITSCREEN or XBOX LIVE and make sure you are using the XBOX live profile that you used to redeem the maps. There will be a new menu option that reads "FREE ZOMBIES LEVEL." Select this option and you will be allowed to download Rezurrection for ZERO points starting on August 23rd. If you still have the code from the Hardened or Prestige Edition but have not redeemed the Zombies Pack, you can use your code whenever you like, Rezurrection will still be free for you when it is available on August 23rd.

You can also acquire Rezurrection for free by re-downloading it from Marketplace or from the STORE in the MP menu on the 23rd.

Everyone:

How do I get my free Moon Theme?

Once Rezurrection has been downloaded, go to the In-game Store in the Multiplayer menus and select the tab marked "FREE OFFER." Presto, free theme.

How do I get my free Call of Duty®: Zombies Soundtrack?

Once Rezurrection has been downloaded, go to the Zombies menu and select "ZOMBIES SOUNDTRACK" from Splitscreen and/or Xbox Live. Make note of your Offer Code in the pop-up window, go to the URL provided and enter your GamerTag and Offer Code. The soundtrack will automatically start to download.

Source: http://www.callofduty.com/intel/492