There are many hosting sites that offer free domain names with their hosting packages. I took advantage of this when I created my web site. There are opportunities to purchase domain names, do a domain name search, and find cheap domain names all online if you want.
If you're looking at creating a web site, I would suggest you do a domain name search to make sure the name you want is available. Many hosting sites have a domain name search function. All you have to do is put in the name you want to search and click to let the search engine check it out for you.
Some people say you should register all of the extensions (i.e., .com, .org ..etc.). And, register all of the possible misspellings. Considering you have to pay for each domain name you register, that could get expensive quickly.
There are sites that will give you a boat load of information on what kind of domain name you should create, and whether it should have underscores, hyphens, one name, and more. I just used my name for my domain name.