Wake-on-Ring (WOR), sometimes referred to as Wake-on-Modem (WOM), is a specification that allows supported computers and devices to "wake up" or turn on from a sleeping, hibernating or "soft off" state (e.g. ACPI state G1 or G2), and begin operation.
The basic premise is that a special signal is sent over phone lines to the computer through its dial-up modem, telling it to fully power-on and begin operation. Common uses were archive databases and BBSes, although hobbyist use was significant.
Fax machines use a similar system, in which they are mostly idle until receiving an incoming fax signal, which spurs operation.