I have a Sony Cyber-shot DSC-H1. It's new and I have hardly used it.
The other day I tried to turn it on and the lens started moving back and forth slightly.
The camera isn't working and shuts itself off.
What is wrong with it, and is it easily fixable?

Because electronic devices like digital cameras are so complicated, it's really hard to know what to do when there's a problem. The first thing to try is taking out the batteries and then putting them back in. If they're low, recharge them or put in a new set and try that. A low battery might cause the problem you're describing. If that doesn't work, you'll need to contact Sony customer service.