Domain Names come in two main forms, generic top level domains (gTLDs) which consist of com/net/org/biz/info/name domains and country code top level domains (ccTLDs) of which there are about 250 including .uk (United Kingdom), .de (Germany) etc.
ICANN (Internet Corporation for Assigned Names and Numbers) is the governing board that regulates the basic workings of the Internet, assigning IP numbers and domain names. For example it was ICANN that authorized the newer gTLDs such as biz and info.
The domain name system consists of registries (the organizations that manage the database for a particular TLD), registrars (organizations that are authorized to register a domain name) and registrants (the slobs like you and me who register them). The almost similiar names are really annoying and confusing but that is what is used.
A seperate registry is generally responsible for each TLD. The main exception is the com/net TLDs which is run by Verisign (because it bought the Network Solutions monopoly) They also used to run org but that transition to another group at the start of 2003. The .cn TLD (for china) is run by the China Internet Network Information Center (CNNIC). Most of the ccTLDs are run by non-profit/academic centers although a few such as .tv, .cc and .ws are run as for-profits. Most ccTLDs have a requirement of some kind of “nexus” with a country (physically being there, registered as a business, etc) but it differs with each one.
By far and away .com is the most popular TLD (~22 Million) vs the next .de (~6 million). It is very easy to find a registrar with which to register a .com domain and they will charge you anywhere from $8.95 to $35 a year to do so. It is perfectally possible to register a domain (at least gTLDs) directly with a registrar without involving an ISP at all.
Generally what you get for that amount is the “rights” to a domain for one year. You can generally register a domain for lengths up to ten years but out-right ownership is not permitted (although defacto ownership/copyright issues will generally prevent anyone from snagging an accidentally deleted domain).
But having the “right” to a domain does not mean that it is “usable”. To make it work as an email address or website there needs to be properly configured nameservers. This is where an ISP or hosting service comes into play although it is possible to setup your own nameserver if you have the know-how. Basically it means getting an entry in a database so that when someone types www.mynewdomain.com into a browser the Internet will know the proper IP address (126.96.36.199) to go to. The Internet could function perfectly well without domain names and rely on IP addresses but we humans would find it a bit more difficult to remember and use.
Some registrars provide a number of services when registering a domain name such as hosting and nameservice…generally you have to pay for this though. My own domains are with gandi.net (the 7 or 8th largest registrar) and they will provide email forwarding and web aliases (which makes the domain name usable) for free. It is not a true nameserver setup tho. If you want to set up a e-commerce website and be taken seriously (not look too much like a small time operation) you need to spring for a hosting service which start at around $100 a year.
Hope this helps…