There is a reason why all movies are shot in 23.976 or simply 24fps - All human beings, no matter what they tell you, can't notice the lag starting from 23.976fps. Fact: All movies and YouTube videos are displayed at 24fps and 30fps respectively.
Therefore, all games, in order to look smooth, must have at least 24fps. This is the VERY MINIMUM for all human beings. If someone can "tell" the difference between 24fps and 120fps, then that person has been brainwashed with the idea of "higher the numbers, better the computer", which is totally false.
Most monitors refresh at 60Hz, or 60fps. That DOES NOT MEAN everything has to be 60fps; it looks the same as 120fps or 24fps. NO ONE, NOT EVEN HARDCORE GAMERS, CAN TELL THE DIFFERENCE!
That being said, it is best for games to display at this rate:
NEVER dips below 24
Average anywhere between 24 and 60
DOES NOT GO ABOVE 60, because at that point your GPU could display more stuff at a reasonable FPS. Ignore this if all settings are on extreme.
Ehh, sorta. If you plug your computer into an HDTV and set the refresh rate to 24 or 30Hz, then move your mouse it will definitely look slow and unnatural. Put it at 60Hz and it will look normal. You really can tell a difference in refresh rates. So ideal FPS would be around 60, which is what most monitors are.