If I purchase a domain name will I also have to pay monthly fee...?

for domain hosting to hold the name? Background: I've created a flash website through the Wix site but I don't like the long name of the site and the fact that it has the wix.com address in there. I want to purchase my own domain name so that I can point the address to my wix website address. After researching different companies for domain names I see that they all offer both domain names and hosting. Do I have to pay for both for the domain name to work? Or will just buying the domain name and then pointing it to my site work?