What's VSync and how does it works?

I recently learned that how I thought vsync worked was wrong, and now knowing the way it really does work, I think it would be worthwhile to make sure everyone here understands it.

What is VSync? VSync stands for Vertical Synchronization. The basic idea is that synchronizes your FPS with your monitor's refresh rate. The purpose is to eliminate something called "tearing". I will describe all these things here.