I have a domain name through Register.com. Do I actually "own" that name or am I merely "renting" it. If I don't renew it at any time, it becomes available to the general public, right?

I also built a website through Register.com. Now, I'd like to have one professionally built. How do I change hosts or have Register.com host my new site and keep the same domain name?

Yes you can host a website elsewhere if you need more space or services.

Register.com is where your domain, dns/nameservers, and hosting is all located. Technically some companies have other companies behind the scenes hosting it all but I'll break it down simple.

DOMAIN:

You can buy a domain from a REGISTRAR which is who you pay yearly for it.

DNS (point to your site):

Then there is DNS and nameserver management (so when you type your domain it goes to your site) which can be handled by the REGISTRAR company (sometimes a third party behind the scenes) or you can personally purchase or obtain for free DNS/nameserver services from another company and manage it yourself. As long as the domain at the REGISTRAR is setup with the third party's NAMESERVERS.

HOSTING:

Lastly you have HOSTING which is where all your ftp and website files are stored and displayed to public when the domain name is pointed to it. This can be handled by yet another company.