In very simple words, domain is your website address/registered name. Someone can visit your site with typing domain name in browser address bar., just like when you write a letter and post in post box, based on the address your letter reaches to the correct destination.
Currently, the Internet Corporation for Assigned Names and Numbers (ICANN) manages the top-level development and architecture of the Internet domain name space. It authorizes domain name registrars, through which domain names may be registered and reassigned.
A domain name consists of one or more parts, technically called labels, that are conventionally concatenated, and delimited by dots, like example.com. Here first part is name for your domain and second part (after dot) is for the purpose identifier of your website. Some of them are “.in”, “.co.in”, “.org”, “.mobi”, “.net” etc. From this you can guess what type of website you are visiting. .in, us, .ca are some of the domains which are country specific top level domains(TLD). For example .in is the top level domain for Indian websites. .org is for organizations and .govt is for government. .com is a identifier for commercial use.
Now I think you might be understood the domain name structure. So you can now choose the domain name which is likely your business name and the identifier(often called extension) which suggest the nature of your business.
There is a different fee for registering domain names, varies based on extensions(like .com, .org .in) and also varies with the registrar. You might be charged from INR300/- to INR2000/- or more per year for domain registration.
There are several domain registrars who can register a domain for you. HostGator, Bluehost, Inmotion, GoDaddy, Namehheap are some of the leading domain registrars. There are many mores but will not be possible to give all the list here.