The most commonly used method to connect to a WAN is through the telephone system. To do so, what piece of equipment would you require?

Modems create an analog signal that carries digital data on it, and strip the digital data from an analog signal that they receive.  That's what's needed to connect through the telephone system, since the telephone system is still one big analog system.

Modem or router -  the device needs to be able to remove the data information from the audio information.