What you do with a registrar:
- you buy the domain name, "mydomain.com" for example: your name appears as the domain owner in the "Whois" database. (You can check it from any registrar, they normally all use the same official Internic one). The price is for one year, the owner has the domain as long as he pays. If a domain was registered and the owner didn't renew its payment it becomes public again and is called "expired".
- you manage the domain name: you say in which servers the domain is managed (you put the server names (dns), for example ns1.myhost.com and ns2.myhost.com, whith or whithout the IP numbers).
- most registrars now offer free basic services on their own servers like email forward or web (http) redirection. It is very useful if you don't have a hosting provider yet.

What to check in their offer:
The first important point is to check that you get the ownership of the domain name = that the domain is registered in your name, not in the name of your provider then let to you. It means that you'll be the "domain holder" or "registrant" in the domain contact details and that you'll be able to transfer your domain to another registrar for the next year.
Registration must include free domain management = you must be able to set and modify any time the DNS settings = the identity of the servers where the site is hosted.
Any price registration under $5 per year for .com, .net or .org is strange.
The big issue afterwards is wether you'll be able to transfer your domain to another resistrar, at what price and if these conditions may change in the future. No registrar that I know set clear conditions for that and we must rely on today's practise...
Customer care: you'll probably deal with a registrar reseller (they tend to offer better prices) so you have to check with whom cutomer care service you'll have to deal = is it managed by the reseller or by the registar. You should start by asking a question before to buy a name (any of the above points for example) and see how quick they answer. But still, you may have a very quick... and very useless answer like Godaddy has been famous for.
Usual ADD-ONS: Forward - Web Mail - Page creation - Hosting
Forward: can be for the web pages requests (sending visitors to your existing home page) or for the emails (forwarding the messages to your existing email address with your internet provider). Most registrars include that in their domain price, some charge you for this "very important highly sophisticated exceptional feature" that should be free. Forward is important for the beginner and for additional domain names (your second domain name or any kind of aliass).
Webmail: is to let you use the email addresses like me@mydomain.com through a webpage (Just like you do with the free address many portals offer and probably the webmail service of your internet provider at home). The only point is that you have "me@mydomain.com" as the "reply to" address in the email you send from this webmail and not "hotmonkey@portal.com" if this was your former webmail address... If you don't use webmail usually and only use your home computer forget it: you'll just change the "reply to" address in your Outlook Express (or mail software) and the forward service will send any message to me@mydomain.com to your personal computer.
Page Creation: these online tools are supposed to be easier to create pages... not too sure. Creating pages offline is very easy and quick. Of course, if you tried to use Microsoft softwares like Front Page, you may be ready to buy a page creation service as you got such a crap result. But otherwise there are many free softwares or sharewares to do basic pages (like pages created with an online page creation tool). If you want an easy and clean software to build a site, download a trial version of Dreamweaver from macromedia.com, it's MUCH easier than Word!
Hosting: For some reason, some people want to have the same registrar and host service, it's a bad idea as packaged offers are always more expensive (but maybe with French Online.net XS). Look at the host prices from Godaddy!!! They are very expensive.

How to choose then?
- Take the cheapest offer "registration + forward" with free transfer OUT (they all advertise free transfer to them, out transfer conditions are usually deep in FAQs or agreements).
- Consider currency conversion charges from your bank when you deal with a foreign company.
- check the registrar's reputation in forums and search engines (do a search with its name or site address).

Cheap: PowerPipe - $8 per domain
Process is clear and straight forward. Not like Godaddy which pushes expensive add-ons. Slow customer service but managing a domain name is very easy. They actually showed how out-transfer conditions were important when they asked a one year registration price for any out transfer in 2003. Everybody shouted and they changed it back to normal. At the end, though I shouted a lot myself, I'm not sure wether it was a pricing decision or just a technical issue when they updated their management page, maybe it was just the "unlock" feature (to enable the out transfer) that has been missing for a while. (Texas)

Hippy: Gandi.net - 12 Euro per domain
Excellent customer service.
Shareholders used to be more interested in free software projects than into profits. (France)

- to be updated -

Resellers:
It is actually very easy to become a reseller and have your "own registration business". (you won't make profits though!)
Two registrars have many resellers: Enom and Godaddy (WildWestDomains).

Domain name business : buying - selling - auction - expired
Domain trade business is dead but there are still a few actors behind the scene. Most of them fight to reregister names that expire though they had existing traffic. This secondary market is the place to make additional profits for registrars and some sort of restricted access auction is going on. It means that a valuable expiring name won't be available for anyone at $9 anymore but sincerly, there is nom oney to make there!
Ressources:
UWhois - Whois.SC - Deleted Domains - AfterNic - SnapNames - DomainState - WebHostingTalk - Ohashi

For domain registration addicted ONLY:
It's possible to buy a domain, meaning the ownership only, whithout any dns management for $5.99 with Enom. It makes sence for VERY FEW situations: you won't know if customers may like it (access impossible), you wouldn't use it for an alias where you should have a forward to your main site. As there are no more free names that could be bought for domain name trade... it may be usefull for professional name holders that have old parked names to renew.
$6 is as far as I know the price all registrar pay to the Internic to register the domain in your name. Only "registered ICAAN" registrar can deal with the Internic. As customers, you have the choice to deal with "ICAAN registrar" like Gandi, Joker or Godaddy, or with resellers like PowerPipe, Online or Cyberwings.