Naturally your ISP isn't going to provide name resolution for your private network. The easiest thing to do on a small network is to use the /etc/hosts file to associate your private network addresses with your LAN computers. Then make sure that the line in the /etc/nsswitch.conf file puts name resolution priority to the local hosts file before dns. The line in that file should look like this.
That line might have other entries in addition to the two that I have listed. The important thing is that the "files" keyword is before the "dns" keyword.
If you have more than let's say 100 servers on your LAN you may want to set up a DNS server. Since you haven't needed one yet you probably still don't need one.