Graphics and video
For a list of games, see games with vertical sync (Vsync) support.
Vertical synchronization is an option used to prevent screen tearing. Screen tearing is a graphical glitch which is perceived as straight horizontal lines across the monitor or as if the whole picture is stitched together by two or more separate pictures. This problem exists because the monitor and the graphics adapter normally works independently, so new frames may not be fully drawn when the monitor display them. Vsync makes the graphics adapter wait for the monitor to signal it's ready for the next frame to ensure all displayed frames are always fully drawn. This has the positive side effect of limiting the amount of frames per second the computer has to draw to the monitor refresh rate (the amount of frames per second the monitor is able to display) which saves resources. Unfortunately Vsync can also increase input lag.
There is none and results will vary between different games, systems and people. If you're bothered by tearing or want the best visual quality, enable Vsync. If you're bothered by input lag or have performance problems, disable Vsync. If you're playing first-person shooters competitively, always disable.
Note that Vsync should not be used only as an FPS limit. If you only want to limit FPS drawn, for example to minimize heat and fan noise, but are not bothered by tearing and don't want to risk input lag, simply limit your FPS to that of your monitor's refresh rate or close to it. An FPS limit is often an option available as an in-game console command or similar.