GoDaddy — This extremely popular registrar (probably the biggest today) offers .com domain names for $9.99 (plus 20 cents) per year (or $6.99 plus 20 cents if you transfer from another registrar). They have a web interface to manage your domains, free web redirection (where people who visit your domain will get transferred to another web address of your choice), free starter web page, free parked page or free "for sale" page, and an optional private domain registration where your domain is registered in the name of a proxy company. Besides .com, they also sell .us, .biz, .info, .net, .org, .ws, .name, .tv, .co.uk, .me.uk and .org.uk, etc. Note that (as with all registrars) the exact price varies depending on which domain you are registering (for example some domain suffixes are more expensive than others). Both credit card and PayPal payments are accepted.
If you're having trouble finding a domain name (whether because of crowding or cyber-squatters), check for a help facility on each registrar's site. Domain registrars typically house search engines that return a listing of available names similar to the one you want. When you search for a domain name at Namecheap, for example, you get both the status of that name and a list of suffixes available for that name. Maybe [Sitename].com isn't available, but [Sitename].biz or .org is.
If you already have a web host, obtain from them the names of their primary and secondary nameservers. Don't worry if you don't understand what these things mean. Just save the information somewhere. The information can usually be obtained from their FAQs or other documentation on their site, usually under a category like "domain name" or "DNS" or "domain name transfer" and the like. If you can't find it, email them. You'll need the information to point your domain name to your website after you buy your domain. Having said that, if you don't have a web host yet, don't worry. Just read on.
A great domain name is a concise, easy to type, and memorable URL that reflects your brand or your website’s subject matter. Avoid using hyphens, strings of numbers, or unnecessary words to make it easy for your visitors to remember and find your website. Remember: A great domain name is one that your visitors can type correctly on their first try.