Jammer1221 is indirectly making a point I was about to make. Since most people don't run their own web servers, you need not only a domain but a hosting service. The easiest (and often cheapest) way to get a domain registered is to choose a hoster and let them handle the registration as part of their service to you (almost all of them do that). I use Netfronts (they aren't as cheap as jammer1221's favorite, but I can say that they do a great job at features, uptime, improvements, and customer care; some hosters won't even answer email - I know firsthand). Anyway, there are tons of hosters out there - you can read about various ones, and find sites who critique them.
Regarding how the domain thing works, I don't know whether you want to call it ownership or rental, but once you have it, it's yours - while it is registered to you, no one can take it away from you. But if you don't keep the registration current (which is cheap but isn't free), it becomes available again. Again, if you go with a good hoster, they will remind you when your registration needs renewing.
). If you need more, you can move up to Silver (250MB and more services) or beyond for a couple dollars more per month. I don't know a lot about the ins and outs of Windows servers (I like Unix/Linux myself), but you can check it out yourself. I'm sure someone else will beat the price (11.95/month for Bronze), but I have had two domains hosted on Netfronts for about three years, and I can say that their quality of service is top-notch - they'll take care of you. They even changed the configuration of their web server for me when I had a need (added additional default page file extensions). They keep their servers upgraded, and normally without any downtime at all. And again, I don't know how Windows servers work, but I have full access to my database for dumps and stuff - if there is something I can't do through the database tools on their "control panel", I can log on with SSH and work from the command line as if I was sitting at the server's console.