By pressing the Calibrate button (in the toolbar) you make sure that the eye tracker is optimised for your eyes. For best accuracy we highly recommend you to do this for each new test participant!
Pressing Fine Tune will launch a software that shows a couple of dots located on the screen. By looking at the dots the software can tell the eye tracker how your eyes are positioned when looking at a certain part of the screen. From that it'll understand where you're looking.
Each dot could be seen as a calibration point. The more calibration points you give the eye tracker, the better calibration you'll get. In Tobii Pro Sprint we use 4 calibration points, which makes a pretty fast calibration. We consider it to be a good balance between accuracy and a quick fine tuning experience for the test participant.