A domain name registration is simply reserving your website's name for a specified period of time. The two parties involved with a domain registration are generally the user who is registering the domain and the domain registrar. The domain registrant is the user who actually registers the domain name and the domain registrar is the company who helps that user with their domain registration.
Sedo is where I bought my domain - over Escrow - and is a global marketplace to buy, sell, and park domain names. They've over 18 million domain names for sale, but the big part is that they're a huge, well-respected company. This may not be important if you're spending $150 on a domain, but in a $15,000+ purchase - which happens in many cases - you will be potentially buying from a foreign entity, that you don't know, and many of them will suggest an escrow payment. This means you put the money in an account owned by a third party that will hold it and only pass it on once the domain has been transferred over into their name (which in this case is what Sedo does). This can be quite disconcerting - but Sedo is well-known and has done many, many deals - and the process is about as full of hand-holding as handing tens or hundreds of thousands of dollars for a domain name can be.
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.