The topic says it all - I encountered the term in a PDF given to me. So ... what does the term/phrase 'War Dialing' mean?
Like the others have said, it is the technique of using a computer to dial down a list of phone numbers in search for other modems or fax machines.
According to Wikipedia, the term was coined due to the technique's usage in the movie WarGames.
It is also the root of the now more well-known term, wardriving, and its derivatives.
Wardriving is the practice of driving around an area with a laptop configured to scan for and collect data regarding wireless networks.
Warwalking is the same as wardriving, but on foot.
Warchalking is the practice of marking your wardriving/warwalking finds with chalk on a nearby wall or sidewalk.
For more information on "warXing", see the Wikipedia article here:
War-dialing is to telephone networks what network-scanning is to computer networks. A computer dials every number in a given number set to map out what's there. In olden days it was looking for unannounced modem gateways into interesting systems, and sometimes fax machines for fax-spam. The robust systems could tell the difference between fax machines, modems, voice-prompt systems, answering machines, voice mail, and actual people, much the same way that nmap can do service and OS identification of network-attached machines.
Other posts have answered the basics of what war dialing is, but here is the reason you do it. A device in your network with a modem attached and reachable from the POTS, is outside the protection of your firewall.
War dialing is part of our security program we do every six months as sometimes a physical modem is necessary for support from an outside vendor. Obviously it should be checked by someone prior to go-live, but I live in the real world and know that doesn't always happen.
There are various commercial (Phonesweep) and opens source (THC-SCan, WarVox)tools out there, it just depend on your needs.
It's using a modem to dial a bunch of numbers to see if you can get a session on another modem or other remote access device. It's the POTS equivalent of doing an NMAP sweep or similar.