Controlling one light from two places uses a pair of switches that have no on and off position at all. Look at a three-way switch and the toggle has no markings, because whether it is up or down means nothing on its own. What matters is which of two paths it has selected.
Each three-way switch has one common terminal, usually a darker screw, and two traveller terminals. Power arrives at the common of the first switch. Two traveller conductors run between the two switches. The common of the second switch feeds the light. Flipping either switch changes which traveller is live, and the lamp turns on whenever both switches happen to have selected the same one.
Adding a third control point means inserting a four-way switch into the middle of the traveller pair. A four-way has four terminals and does one job, which is to either pass the two travellers straight through or cross them over. Every additional control point is another four-way inserted into the same pair, which is how hotel corridors end up with six switches for one run of lights.
This layout is why replacing a three-way with an ordinary switch never works, and why smart switches complicate things. A smart device usually needs constant power, which means it needs a neutral conductor in the box. In older three-way circuits the neutral often sits only in the box that feeds the fixture, not in the remote switch box, and there is nothing to connect to.
Troubleshooting follows the same logic. If a pair of switches works from one position but not from a particular combination, the common and traveller wires have been swapped at one end. If the light works from one switch only, one of the travellers is broken or landed wrongly.
Understanding the traveller idea also explains a common annoyance in older houses, which is a light that responds to one switch only if the other switch is left in a particular position. That is not a fault in the fixture. It is a wiring mistake at a terminal.
When rewiring or adding control points, pulling a neutral into every switch box is worth doing even if nothing needs it today. Almost every device that might replace a plain switch later, whether a timer, an occupancy sensor or a connected switch, will want one, and running it during open-wall work costs nothing extra.
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