Purchasing a domain is one of the first steps in controlling and owning your identity on the web.
Domain registrars handle the reservation of domain names and assignment of an associated IP address but do not themselves manage domain names. A separate organisation known as a domain name registry undertakes this management. Depending where you reside and the purposes of your website then you may have a greater or more limited choice of top-level domains (TLD) which you can buy.
After registering a domain name
Once you've purchase a domain name then you need to link this up with some hosting unless your hosting provider and purchased package includes domain registration. If you haven't already got [hosting] and unclear about this then consider looking into self hosting or shared/managed hosting.
Your domain registrar should provide a section in your account settings to specify the details of your hosting knows as DNS settings. This includes a group of records which you should be able to obtain from your host to specify the numerical address for your domain to point to.