First, there is no such thing as a two-way switch.
( Single phase is another term that messes people up. It has, in common parlance, two phases! )

Household switches come in single-pole, three-way and four-way. A single-pole switch is used when something is switched from only one location. To switch something from two locations, two three-way switches are used. For switching from more than two locations, two three-way switches and as many four-way switches as required are used.

Why they are called three-way switches when they are commonly used to control a light from two locations may be lost to antiquity, but it probably has to do with how these switches are wired rather than how they are used.

On the back of a three-way switch you will find three connection points, typically screws. One of them is the “common”, typically indicated by a black screw. The common on one of the three-way switches gets connected to the hot wire of the circuit. The common on the other three-way gets connected to the switch-lag, the wire to the light that is hot when the light is on. The other two screws on the first three-way switch get connected to the corresponding two screws on the second three-way switch by wires called “travellers”. Which traveller goes to which screw doesn't matter. The three-way switch opens or closes the circuit by changing which of its travellers is connected through the switch to common.

The actual wiring to achieve these connections can be done in many ways and even a lot of electricians only learn one or two ways of wiring a three-way system, then repeat by rote. Of course repeating by rote doesn't help when troubleshooting anything different. Like many things, an understanding of the underlying basics can make all the difference. Fortunately when a 3-way is messed up it usually isn't dangerous, it just doesn't work.

The diagram below demonstrates what happens inside the switches in a simple 3-way switch arrangement.

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