What happens if you wire a 3-way switch incorrectly?

Depends on what you do wrong. Worst mistake would be hooking the hot wire to a ground screw. If you did that without a functioning ground wire you could get a bad shock—and the light wouldnt work. Most likely case for a mistake is in a three way system where the common and a “traveller” gets mixed up.

What is the purpose of 14 3 wire?

A 14/3 electrical wire means that it is 14-gauge in size and it has three primary wires plus a ground wire. The three primary wires are red (hot), black (hot) and white (common). This wire is used as two 120-volt circuits that share a common return wire.