I'll hopefully end up writing dozens of posts on health insurance and how the healthcare system in the US (well, California) differs from the NHS, but let me start with this:
We're with Kaiser Permanente. It's called a "HMO", and an HMO is fairly similar to the NHS.
You pay yer money every month. (Mostly, my employer pays. It's, like, $8k a year or something for the family. Yikes.) You have a doctor, they're exactly like your GP. If you need something, your GP is the person you go to. If you try to bypass your GP, you'll have to pay for it totally yourself.
Of course, if you have to go to the ER ('cos that's what they call it here!), that'll be covered. Probably. We haven't had to yet. If we did, we'd probably go to the Kaiser ER, since it's not far away from our house.
In the UK, of course, the NHS is pretty much free, and you won't have any nasty financial surprises. I really love the NHS, I reckon it's one of the greatest things the British have ever built, possibly *the* greatest. It definitely has its flaws, but it's still amazing.