For console gaming the framerate is largely static. You don't need to worry about adjusting settings for a better framerate with console games.
For PC gaming however, depending on your hardware and your settings, you'll get varying framerates. A higher amount is better, and it generally affects how fluid the game looks. Besides higher framerates just looking better, if you have poor hardware or your settings too high, you'll likely not just get poor framerates, but fluctuating ones as well (as in, it won't run at a steady 30 FPS but instead shoot around between 5 and 40). This is very jarring.
EDIT: to answer your question a little more, different games don't need more FPS than others. 60 FPS is solid universally. However, some games may be more hardware intensive and as a result require a better machine to get the same amount of FPS as you would in other games.