FPS = Frames Per Second, or how many images are flashing across your monitor (hopefully, but thats a much more complex topic) per second. Generally, you want this to be as high as possible, up to a maximum of your monitors refresh rate (most of the time, 60FPS or Hz).
Your typical movie plays at 23.97 (I think) or 24FPS, though generally 30FPS is considered minimum for smooth gaming. Your more graphically intensive console games tend to run at ~30FPS, while your twitch shooters (COD) run at 60FPS. I personally notice an FPS under 45 and will reduce settings to keep it above that, so its largely personal what your threshold is.
your FPS is determined by many factors, but it mainly boils down to how well your CPU and GPU perform. Your CPU, once you cross a threshold (between an i3 and i5 level of performance) having a stronger chip just simply wont benefit you as you have all the CPU grunt you need. The GPU does benefit from having more grunt, the more powerful it is the greater settings you can enable and still get a high FPS.
Internet speed doesnt play affect FPS, or at least not directly. If your in a multiplayer game and your connection is struggling, everything will jump around as, but your FPS will still remain high. So in this case its almost your having multiple frames dedicated to the same image being displayed.
so what fps will i get on bf3?
Facebook
Twitter
Instagram