Standalone machines will generally not use a DNS, unless they connect to a larger network once in a while. Even then, they may use a DNS on this larger network.
Linuxconf checks at different times (boot time for one) for DNS connectivity. If the DNS fails to answer after a few seconds, Linuxconf will complain and allow you to skip network configuration. A bad or faulty DNS configuration is very dramatic (they often give the impression that the computer is stopped and unresponsive).
This test is helpful, except when the machine does not have full time access to a DNS. The checkbox is just there to tell Linuxconf not to care much about the DNS. Other than this, it won't change the operation of the machine.
You generally enter the domain you use most, usually the domain of your organization.
You must enter IP numbers, and not machine names. For a standalone which never connects to any larger network, this is optional.
An alternative to the default domain is the "Search domain" field. While this adds some overhead to DNS searches, it may be a good time saver if you talk regularly to several machines spread over different domains.
Note that using "Search domains" is exclusive to the "default domain" field. Use either one or the other.