Just go to any domain registrar, find a domain extension you want to purchase (such as .com, .us, .net, or whatever else), and it’ll give you the option to input the name of your website. Once you input your name, it will search for whether or not the domain is available and notify you of it.
Conversely, just check whether or not the website is currently active. If you want to go for something like “somethingsomething.com”, you open that website through your chosen internet browser, and it might end up showing you an actual website, or something like this:
If you want to go the registrar route, my suggestion is Nominus.