How do 3 way switches work and how do you wire them ?

See photo, it shows the simplest method. There are others with feed coming into any of the devices, but need to be knowledgeable of wiring to do other methods.

The inside of a 3 way switch is nothing more than a single pole double throw switch.

Here is my way of thinking.

Rules:
Three way switches work in pairs. You must have two and only two. You can have a four way switch in between a pair of three ways but you only have two three way switches.

Three way switches do not have an on or off position. A pair of three way switches are either in the same position (electrically) or they are not. Being in the same position amounts to "ON". Not being in th same position amounts to "OFF".

A three way switch has one common terminal and two traveler terminals. The common terminal is always connected to one or the other of the two traveler terminals.

Not all three way switches are built the same. The common terminal terminal is not always in the same physical location. The common terminal is always a different color than the traveler terminals.

There is always three conductors between two three way switches. The third conductor is sometimes a neutral conductor and sometimes a hot conductor. There is one situation where half of it is neutral and the other half is hot. The third conductor is not shown in the logic diagram.

There are five different ways of wiring a pair of three way switches.

Fifth way