Alright, first FPS can mean two things. Sometimes, it means first-person-shooter, the games where you only see your hand holding a gun. You're looking from the perspective of the character, through his/her eyes.
Now, to explain frame per second in short, replace the word 'frame' with 'pictures' per second and you'll get a better idea.
Basically, animation consists of a sequence of images displayed very quickly to give the illusion of motion, kind of like when you draw stick figures at the corner of a book, and you flip between the two pages and it looks like he's moving. Why's this relevant? Well, the fps count or 'frame-rate' dictates how smooth the movement will be. A lower framerate will look jittery, whereas a higher frame-rate will look very smooth.
Low fps = a slide show (ex. 0.1 fps)
Fluid motion (movies/videos) = 24 fps
Video games = 60+ fps
in video games, a higher fps is very important because the player is immersed into the game and needs to be able to react smoothly and quickly without being limited by any jerkiness, whereas something as passive as watching a movie doesn't need as much of a higher fps, enough so that motion is smooth enough
so think about this, a graphics card needs to create the image, so the higher you set the visual quality, the more it's going to stress the graphics card, therefore, it can't produce the frames as quickly, meaning a lower framerate, and that's why you'd need a stronger card.
any questions?