Copyright law extends to all creative works, including video games. Copyright law touches many aspects of a game, but not necessarily everything. Only the expression of an idea is protected by copyright.
If a video game is copyrighted, many elements are protected including the code as a whole, the dialogue, audio and video recordings (including an individual recording gameplay), and storyline of the game.
What's Not Protected
The idea of the game is not protected under copyright--only the expression of the idea. Also not protected under copyright are the game mechanics so long as its common to a particular gaming genre. (The mechanics of first person shooters, for example, are not protected under copyright).
Copyright v. Patents
Patents require novelty. It must be a new and different way of doing something. Most games run on similar coding and mechanics so its rare to come across video game patents because of this necessary element.
Copyrights, alternatively, protect the expression of an idea through design, both from a technical and artistic perspective. Copyrights are much easier to obtain, and protect a video game for much longer than a patent would.
How to File
Head to the United States Copyright Office website and register the game you've created. For a nominal fee, you can protect the game if someone else takes credit for the work. Be sure to put a copyright notice and date on the game as well.
Emerging Legal Issues with Video Games
Even the gameplay of a video game is protected by copyright law. Posting video of personal gameplay online is a copyright infringement. If asked to remove the video (known as a "cease and desist"), do so, or risk a copyright lawsuit.