Best definition
Using a computer to dial telephone numbers within a given range, usually with the intention of finding a modem carrier signal. The practice largely predated the widespread penetration of broadband Internet connectivity; at the time, many businesses, agencies, and individuals operated computer systems “on-demand” through telephone-based modems, each of which might (or might not) offer a unique (and possibly privileged) selection of information, as well as possibly offering access to powerful hardware or a platform for reaching other networks and systems. Usually, the wardialer would be covertly planted on a public, shared, or corporate phone line, left to operate for a limited time, then retrieved so that any “positives” (phone lines returning a modem carrier signal) could be investigated later from yet another location. The practice often went hand-in-hand with phreaking, for obvious reasons.
Today, some telemarketing and social research firms use similar programs (usually working from a digital phone book) to reach residential numbers in search of sales or social information. Also, on rare occasions, people engaged in social engineering have used a form of this process to explore “gaps” in corporate phone listings to discover (and identify the owners of) unlisted numbers.
This term directly inspired the term wardriving, due to similarities between the two practices: both return unpredictable results, both require real-world travel, and both activities are done for rather similar reasons. On the other hand, while wardriving is inherently focused on and limited to a specific geographic area, wardialing is a prototypical bruteforce process, much like password cracking, and can theoretically be achieved from any location with a dial tone.
In the 1983 movie Wargames, a teenager engages in wardialing and discovers a backdoor into the NORAD (NAADS) computer system. He then accidentally runs a simulation which almost turns into World War III.

The wardialer is dead. Long live the wardriver.

wardialing: define #2
Used mostly in the ’80s and ’90s: having your computer repeatedly dial a number to try and get into a limited-space modem pool immediately after another user hangs up.

An identical technique was sometimes used to get the first call for prizes in radio “call-in” shows, thus leading to the adoption of random “fifth caller,” “seventeenth caller” etc. by radio stations to circumvent this practice.

“Dude, AOL was so busy last nite i was wardialing!!!1!!” or: “WKRX still gives the prize to the first caller, so I wardialied it all yesterday and got this sweet CD”.
wardialing: define #3
war dialing
War dialing is scanning a range of telephones (ex. 555-555-xxxx) to find all sorts of stuff, like fax machines, carriers, and BBSes (remember this was a loooong time ago)!
Joe: Hey, I was war dialing our area code and found a hacking BBS!

Bob: Sweet! What’s the number?

wardialing: define #4
war dialing
Using a port scanner to find vulnerable computer systems.
“I was war dialing the school’s network and found a backdoor, so I gave myself strait A’s.”