As discussed on our Getting Started with Domain Names page, when you "get" a domain name, you're technically licensing it from a registry. You have control over it, consistent with that license agreement.
There are many ways of acquiring that domain name, including:
When you acquire the domain name, you will want it pushed into an account in your name at the registrar where it is held (i.e., you and the seller both have free acounts at GoDaddy... just push it between accounts). Alternatively, you can request the EPP keys for the domain name and once you have that, you can push the domain name to any account at any registry (as long as the domain is available for transfer, not expired and not "locked" at the registrar). Check with the individual registrar for details on this.