Domain registration is the process of associating a name with your website. Every site on the Internet is actually a numeric address like this site’s numeric IP is 188.8.131.52 but the domain name associated with that is BareTechnologies.com. Both those address get you to the same page, but 184.108.40.206 is difficult to remember and thus why you want a domain name associated with that. You don’t have to have a domain name as you could use the numeric IP address, but there is no question that it’s worth it to pay a small fee to get a domain name which people might remember.
Registering a domain name generally would cost about $20 a year, but there are now reputable companies offering this service for less. Either way, this will be a yearly fee and is separate than your hosting fee. I highly recommend Hover as this is the company I use for most of my domain names. Their prices are reasonable, and their service and features are all top notch. I got switched to them for some of my domain names when the company I was using got bought out, and at first I was upset about the change, but I soon discovered the advantages and I’ve since been slowly moving my other domains there.
You might have noticed that I saved this step until after you found a host for your site as many hosting companies will provide domain registration for you which will save you this step. Some may provide it for free, but most will charge you a fee. If paying, make sure to find out how much as some companies charge significantly more than you’ll have to pay if you registered it yourself. It’s often nice if the host company does this as they’ll probably setup everything for you as well saving you the hassle, but it’s not that difficult and if you need help with this step just ask.