Suppose I have a server in my network that would like to talk to the database server.
How would it talk to the DNS, and then point to the database server?
You seem to be unsure of how DNS works, based on your question here and I'm assuming the similar ones over on Stack Overflow.
Every computer on your network will need to be configured to talk to a "recursive" DNS server that can it can ask to look up entries on its behalf. All recursive nameservers come with a a list of "root" domain servers to find arbitrary domains, but you can override their lookup for certain domains for various reasons - a common example is ".corp".
Your recursive nameserver, then, must be told what the "authoritative" DNS servers are for certain local domains. With some DNS servers this is very simple, with others you'll have to run authoritative dns on a different IP (djbdns).
You do not need GoDaddy for internal domain names local to your network.