I will answer regarding worldwide top level domains .com/.net/.org and give a bit of information about country specific domains afterward. I will keep this to the very basics to set up a domain, email and a website. (There is a lot more that can be added if you need to do more fancy things)
When you buy a domain the first service that your require is a Domain Name Server (DNS).
The domain name server is the first port of call for a request to your domain and all requests for your domain will be routed via internet central (the core .com/.net/.org servers) to your domain name server.
The job of the domain name server is to route requests for hosts within your domain to the appropriate place.
The domain name server holds a DNS record, which provides mail server records (MX) records and host records (A) records.
Next you will need a mail server to handle email sent to your domain.
Your mail server records will tell them where your mail server is - the mail server will route/hold all mail addressed '@example.com' depending on how it is set up. (Many domain resellers will provide you with a mail server and email accounts if you don't want to do anything too fancy.)
Finally you will need a webserver to host your website. This webserver is listed in the 'A' records of your domain name server. The DNS record contains a record of the form
www A 123.123.123.123
where the 123.123.123.123 is the IP address of your web server.
There are many hosting companies who you can get to host your site and they will be able to supply the IP address of your webspace, if they don't provide a complete package for you.
That's about it there isn't anything particularly complicated or highly technical to worry about.
Regarding country specific domain names - each country will have its own way of handling the registrations but in most cases they follow a similar pattern to the .com/.net/.org domains. There is a central authority which will appoint registrars who can resell domains to the general public.
The domain name servers and DNS records work in the same way as .com/.net/.org except that requests will be routed from a national core server (note there is usually more than one.)
You might find this helpful - [icann.org...]