How exactly does DNS work?

Solution 1:

Hostmonster wants to be come the new authority for your DNS zone.

You DNS zone is equivalent to your domain name example.com. In that zone file is defined what IP address example.com points to (as well as all sub-domains, like www.example.com).

For every DNS zone, you must define a name server that handles that zone. So far, that name server was provided by name.com. Hostmonster has now provided you with 2 new name servers for your domain.

To illustrate this, let's request the whois for superuser.com:

$ whois superuser.com

Whois Server Version 2.0

Domain names in the .com and .net domains can now be registered
with many different competing registrars. Go to http://www.internic.net
for detailed information.

   Domain Name: SUPERUSER.COM
   Registrar: NAME.COM LLC
   Whois Server: whois.name.com
   Referral URL: http://www.name.com
   Name Server: NS1.SERVERFAULT.COM
   Name Server: NS2.SERVERFAULT.COM
   Name Server: NS3.SERVERFAULT.COM
   Status: clientTransferProhibited
   Updated Date: 03-feb-2012
   Creation Date: 01-nov-1995
   Expiration Date: 31-oct-2016

As you can see, it lists 3 name servers which are responsible for that domain. These are the name servers responsible for all names in that zone. They are the name server entries that are supposed to be replaced in the process you mention.