How do I find out if my computer's hardware is compatible with Ubuntu?

The Linux kernel supports a vast range of hardware. Of course, some things are better supported than others. Chances are your machine should run alright regardless. If you've got atypical hardware, or a strange environment, then that could add complexity. I've found some wireless network cards are still not well supported and getting great video performance is fairly dependent on your video card as some modern cards can require some work to get running and others work with next to no thought.

Your best quick test is to download and burn the Ubuntu installation CD. Then boot up the Live version of it from disc. This doesn't touch your hard drive. If it runs off CD fairly well, then you should expect it'll run off your hard drive reasonably well should you install it to.

Alternatively, you can go do a bunch of research in to supported hardware. But just trying it is a good option in your case I think.