Registrars are the shops where you buy domain names. They deal with the organisations that run each ending, such as Nominet for .uk addresses, and give you an account where you manage renewals and settings.
Your domain can be registered with one company and your website hosted with another. Many people end up with both in the same place for convenience, but it is not required.
Prices for the first year are often low, with renewals costing more. It is worth checking the renewal price before you buy, and whether the registrar charges to move the domain elsewhere.
Whoever controls the registrar account controls your web address and your email. If you do not know which registrar you use or cannot log in, sort that out now, while things are working, rather than when your email suddenly stops.