There are a couple of tutorials on the wiki for installing on Ubuntu and they have a DNS tutorial in them, they are describing an installation of BIND on the Zimbra server itself. There's also some details in the forums (second post in that thread) of what's required for a DNS install on another server in your LAN. If you want some more detail on DNS (but stil fairly readable) then this site has a good documentation.
The important thing to remember is that your server requires a FQDN (Fully Qualified Domain Name) that consists of a server name + a domain name so that: mail.domain.com would be the FQDN & domain.com would be the domain name in your DNS records. Your hosts file should look like this:
and should not contain anything else. Your resolv.conf should only contain the IP address of your DNS server and the DNS records should be set acording to one of the tutorials and to reflect your domain name and Zimbra server IP.
127.0.0.1 localhost.localdomain localhost
192.168.1.10 mail.domain.com mail
When you set-up your DNS you can test it with the following commands:
Obviously the first two commands just list your configuration files, the last three commands display information about the DNS resolution for your server. WHen you've done that and you're happy it's OK (post those details here first, if you want confirmation) then I'd suggest you uninstall Zimbra with the following command:
dig yourdomain.com any
dig yourdomain.com mx
host `hostname` <-- use that exact command with backticks not single quotes
That will remove everything in the /opt/zimbra directory structure. You can then install zimbra again and when you get to the question about DNS records and 'do you want to change the domain name' you should answer yes - it will show the FQDN of the server and you should enter just the domain name: domain.com Apply the changes and let the install finish.
That should be about it, let us know how you get on.