Go to the domain registration service website, input your desired name and choose your favourite tld (eg. .com, .net, .ca etc) and search. The system will let you know whether the domain is available for register or not. After which, you can proceed to add the new domain to the shopping cart and choose your 'hosting' plan along with it.
Your own website should be online within 24 hours(provided that correct nameservers are apply to the domain)