Great question! If the moon suddenly disappeared (that would take some doing!) firstly there would be no tides, bad news for the surfers and for the beach wildlife which relies so heavily on the to and fro of the tide! On that note also the animals which rely on the phases of the moon would also be in trouble.
Secondly the Earth would speed up its spinning rate and cause the days to become shorter. It slows the Earth’s spin now due to a process called tidal torque. The moon used to be much much closer shortly after its formation, and its tidal forces were huge back then, so much so that tides might not have been just water waves, but the very surface of the Earth itself! Its been moving away from the Earth gradually over time, at a rate of a few centimetres a year. There will be a point where we won’t have total eclipses again and the moon won’t be part of the Earth system. That though will take tens of billions of years!