you better contact your hosting provider, they will help you better. BTW: who is the hosting provider??
buy at a domain registrar like namecheap or name.com or moniker instead of buying from your hosting.. i recommend you do that.
Yeah, simply visit hostgator and then register. Their hosting package already includes a domain name.
Unless you already have the domain name, in which case, you simply go to your domain account and change the DNS to your web host name server. I write this obviously assuming you have a hosting account somewhere and you purchased a domain name through someone else. Otherwise, see above comments, and I will shut my mouth and sit in the corner.