How do I set up an Ubuntu server as a secondary DNS

I assume you are using BIND, which you wrote in a comment. First of all you need to copy some files to your secondary name server:

• named.conf
• named.local
• db.cache

Now open the named.conf at your secondary server and change some values:

zone "example.org" IN {
  type slave;
  file "bak.example.org";
  masters { 192.0.2.34;};
};

The type is now set to slave, so BIND knows this one is no master. The *file*name at your master server is probably something like db.example.org. At the secondary it is changed to bak.example.org. However you can select a na e you like. I like it this way because it makes clear, that this one is just a backup nameserver. The last entry is the IP address of the master nameserver. BIND will contact it and retrieve the zone files from there. So you do not need to download those files manually.