Domain registration and hosting are two separate services. Many companies offer both, but they are completely different services. Here’s a metaphor that may help you understand the difference (it confuses a lot of people):
Think of a hosting company like an apartment you rent, a place where your web site can live.
Think of the domain name like your phone number, one that you can transfer (or forward) to a new apartment if you ever needed to do so.
Some web hosting services, such as Weebly, including website building tools and templates with their hosting so you get everything you need in one place. You could think of that like getting a furnished apartment.
If you create a custom web site with a program like Dreamweaver or WordPress, you’ll need a web hosting service that doesn’t restrict you to their site building tools.
Web hosting varies the way apartments do. When you sign up for web hosting at a new company like Dreamhost or Hostgator, you get a cheap apartment, but one it’s still one that you can furnish yourself. When you sign up with a company like Rackspace.com, you got a much nicer apartment, but you pay for the view (or in this case, the faster, more reliable service).
If you use WordPress to run your site, consider using a dedicated WordPress hosting service, like WPEngine, which is optimized to run WordPress faster than general hosting service.
These and many other web hosting services let you upload your own web site to their web server or install a WordPress blog, much like you would furnish your own apartment and hang art on the walls.
You can use the same company to register your domain name and host your web site, or you can use two different companies. If you use two different companies, then you then have to essentially transfer your domain from your domain registrar to your web host, much like you might ask the phone company to move your a phone number to your new apartment. In the case of a domain, however, it’s usually called changing DNS. If you ever want to change the company where you registered your domain, that’s kind of like changing phone companies while keeping the same phone number and it’s a bit more complicated than just changing DNS.
For more tips on how (and where) to register a domain name, check out the article: Can Anyone Register a Domain Name?
For more tips on web hosting, check out the article: Choosing the Best Web Hosting Service.