Basically the refresh rate of the monitor is the maximum FPS visible. So using a 60Hz monitor, you will only ever actually see 60FPS, even if you're pushing 60+ FPS.
The monitor will never affect game performance, but in order to take advantage of a 144Hz monitor(i.e. actually seeing 144FPS), you will need a video card powerful enough to actually push 144FPS. In order to do that in some games, you may need to turn some settings down.
Watch this Techquickie for an in depth explanation: