Depending on countries, the common video frame rates are 24 FPS, 25 FPS or 30 FPS. Videos shot in these frame rates are considered to have the normal speed.
Higher FPS makes the video appear smoother, when playing the video in real speed.
Videos shot in 60 FPS or in higher frame rate are often created for a purpose to make a slow motion videos. These videos are shot in high FPS and then they are played at a normal speed.
Whether a high FPS videos have a higher quality depends on ability of the camera sensor to shoot a high FPS videos. Not all the camera sensors are able to shoot a video in high FPS, but the camera can create a slow motion video, using a software - frame interpolation.
High FPS videos created by a frame interpolation have a lower quality than a high FPS videos created by a camera sensor.