A domain name and "URL" refer to the same thing, conceptually. Both terms refer to a selectable name. Domain names work similar to the license plat on your car. Let's say your car is a website. Then your license plate is what gives you "authorization" to run on the road. Technically, you could run your website without one, but it would be a drab world of cryptic numbers, dots, and weird names that mean very little. So, OK, it's not a perfect analogy. Let's say it is more like the Vanity Plate world. So now, if your want to call your car "Rubber Duck" ... oops, too many letters ... how about RUBRDUK ... you can do it. The main difference here is that there is one secretary of state office for the entire planet. So if your name is taken ... too bad. Except for one caveat.
That is something called Top Level Domain, or TLD. A TLD refers to the last letters that follow the period (dot, in the lingo of the net). For example .com (dot-com) is a TLD. So it dot-net, and dot-org, and dot-edu. Some TLD's are reserved. For example, you cannot just buy a dot-edu. You must be a certified college or university. Same with dot-gov. You must be a governmental body. But most TLD's are freely available for you to buy. So if your name in dot-com form is taken, you can try a different TLD.
Here are some common TLD's
When you choose a domain name for your site there are some generally accepted best practices that will help you a great deal.
Once you have a domain, you can use something called a sub-domain to continue the branding. For example, if you own the domain ABC.com (don't you wish?) you might point www.abc.com at your general information site, and blog.abc.com at your blog, and service.abc.com at your customer service site. The first part, "www", "blog", or "service" would be considered a "sub-domain". Some domain name provides and hosting companies allow you to have sub-domains at a small extra charge, or even for free.