Might be a bit wrong but I think it works like this:
Registring the domain
You register a domain at some registrar. They do have some kind of contracts with the operators of the root servers. For .com domains that's NetSol.
So, basically when you register a domain name you also need to enter the nameservers. E.g. mydomain.com uses as nameservers ns1.myhost.com / ns2.myhost.com.
If the nameservers of ns1/ns2.myhost.com don't exist, then glue records are needed. This means that the owner of myhost.com tells the registrar that it's a glue records and that those nameservers can be found on specific IP addresses. E.g. 18.104.22.168 and 22.214.171.124 (normally two different IPs are required for the two different nameservers)
All of this information is then added to the root server.
Making a dns request
Now in your browser you want to surf to mydomain.com. If the domain is also in your dns cache, then first it will check whether the information has expired (most domains are set to about 48h... after that the information will be renewed again by checking on the domain info). If it has not expired then the IP of the actual server where the domain will point to will be retrieved from the cache and you will be directed there.
At this point apache (or another webserver or ftp server or ....) comes into play. Apache on the server gets a call for the domain "mydomain.com". It checks its configuration whether it has an entry and if so, then it will display the according pages. If not you get an error.
However if the domain info has expired in your cache or if there is no information yet in your cache your computer asks then (normally) your ISPs dns server whether this one has temporarily saved the domain info. If not then it will go up the ladder (I don't know how many steps there are) until you arrive at the root server. If the domain exists then the root server will have an entry and all ladders in between will put the info into the cache... so that not for every request the root server are being queried.
At least I think it works like that... or closely like that.