Pausing a line means that you will stop being prompted to review it. You can always choose to unpause it and start reviewing it again.

This is useful if you have decided to stop playing the line and do not want to be quizzed on it anymore, or if you have two equivalent alternatives and need to pick one to play (to stop confusion), for instance, if a course recommends both 5. Nf3 and 5.Bc5, both good moves in the same position, but to build a course you should stick with one (especially at non-master level!).

The pause button will only be visible if you have learned (seen the line) at least once. Otherwise, there just isn't anything to pause!