What internships should an aspiring video game programmer apply for?

Yes, the short answer is "go make some games" -- but that's a tricky answer, too. if your goal is to be a game programmer, then start small and make your own copies of Pong, Breakout, Tetris . . . things like that. Once you master those you can try to move on to an original small game or try to reproduce a more complex game like a Mario or Zelda clone. This way you have a working model to compare to and know if you are doing right or not.

The bigger answer is to look around you and see if there are other indie game developers around. Check Meetup.com or IGDA.org to see if folks meet regularly to compare games and help each other out. In Dallas, we have the Dallas Society of Play which is a group of indie developers working mostly in the Unity 3D engine. That's how you get real experience. For the most part, there are exceptions, internships are rare in the digital game industry. No one has the time to walk an intern around and no one is going to hand off important work on a $5 million project to an intern. If you happen to leave near a game studio and talk your way into an internship, great. But don't hold your breath. Working with an indie community in the evenings and weekends will get you the same experiences and is easier to find.

I used to do some freelance writing for a few video game websites, and all the game developers I have met all stressed the importance of building a portfolio of work on their own. Even an internship opening will require you to demonstrate some basic skills. The good news is that there are a ton of options that are cheap for people wanting to learn game design/development now. Unity is a free game engine. GameMaker is a good tool for learners. Find free tools like those, scour their forums and communities that discuss how best to learn/use them, and then just start making games!

Whether you're aiming for an internship or a long-term career in the games industry, making as many games as you can is essential.