I am searching for domain names and started wondering where these actually come from? Can anyone set up a .something domain name?
Irmguard, setting up a web server and having a domain are two separate things. Typically, you can either rent server space (called "having a server hosted at ") or have your own server (done by many businesses, but you could even turn your PC into a server if you wanted to).
Having a domain name, on the other hand, is a matter of registering it with a registrar, and then associating it with the IP address (in numbers, e.g. 18.104.22.168) of your server. It's possible to give a real server more than one domain name (mult-ihoming).
BTW, although we often talk of "owning" a domain name, I'm not sure that anyone actually "owns" it in the normal sense of the word. It's more like having a patent - you're granted permission to use it, and others are prevented from using it, for the period of registration. The registrar doesn't own it beforehand or when your lease expires.
If you have a particular project in mind, it may be useful to tell us what you want to DO, and we'll be able to give you more specific advice.
Anyone who wishes to buy a domain name can do so as long as someone else hasn't beaten them to the punch. All you need to do is find a local registrar such as Network Solutions or GoDaddy in the US, Marcaria or EuropeRegistry in Europe, find an available domain name that you like and pay a fee for them to set it up. There are restrictions to what domain names can be used such as an educational institution for .edu addresses, a government entity for .gov addresses, etc. Until a domain name is claimed, nobody "owns" it. In reality, nobody really owns any domain name; they are leased for a period of time (generally 1 to 3 years) and require re-registration or extension for continued use of the domain.
Tell me more about the registrars. Can anybody set up a web server and lease out domain names? Does a registrar run a web server and this way create new domain names? I really appreciate your help and time.
Anyone can run their own web server or have a website hosted by another company and lease domain names for it. All registrars that I know of have their own web presence and will allow online transactions to create and "buy" your domain name(s). If you are using a hoster for your server, for a small fee many will also take care of the domain registration and configuration of the proper DNS records for you.
A slight correction to what Oron said above: a domain name does not have to be associated with an IP address. For example, there are many companies that have purchased domain names where the name alone is derogatory toward the company. These domain names don't point at anything at all. During the late 90s and early in 2000, people were scarfing up these names and using them to badmouth the companies they referred to, often by making a duplicate of the company's official site and then modifying it to project a negative image. So not all domains, even though they have been purchased, will get you to a website.
Most people after purchasing a domain name will want to have it associated with the IP address of their web server by having an A (and possibly AAAA) record created in the DNS system so their domain name will resolve to their server's IP address. Some will also want to configure CNAME and/or DNAME records as well and if they will be hosting their own email server for the domain, the MX record is also needed.
As far as control of domain names and the DNS system in general, that is the responsiblity of ICANN. They are the ones who certify registrars and maintain the TLDs (Top Level Domains, such as .com, .edu, .gov, etc). They also oversee IANA which is the organization which controls all of the DNS root zone. IANA coordinates with the regional IRs (Internet Registry) for each area. They are RIPE NCC for Europe, APNIC for Asia and the Pacific region and INTERNIC for North America and all other regions not covered by their own specific IR. These are the groups that the individual registrars are working with during the creation or removal of domain names.
For more information regarding domain names and an overview of how the various organizations work in order to provide this service you should refer the the RFC (Request for Comments) at IETF (Internet Engineering Task Force). It is located here http://tools.ietf.org/html/rfc1591