Ok so I take it that your asking what would happen if the moon disappeared and not what would happen if we never had a moon from the start. Well, alot of things would actually happen and guess what, I'm going to list them out for you;
1. No more tides: The gravitational pull of the moon on the earth is what causes ocean tides and without it, tides would become a thing of the past. A significant number of sea life depend on these tides to supply them with food and resources to survive so a lot of species of fish would undoubtedly go extinct.
2. The earth won't handle change well: If you had a friend who walked with you to school everyday and sat by you in class all your life and then they just disappear, it's bound to cause some serious issues for you. The earth is no different and since the moon has been with it since the very beginning it's going to have some issues. Volcanoes may erupt, earthquakes may occur, who knows.
3. The earth would spin faster: The gravitational pull of the moon actually slows the earth down a bit with a phenomenon known as tidal friction. Without it the earth would spin just a little faster and sooner or later, all our clocks and watches would be telling the wrong time. Days as we know it would no longer be 24hours.
4. The stars would become more visible: The light from the moon actually prevents us from seeing the stars, similar to how you can't see anything behind a torchlight pointed at your face. The stars would become so much more visible and the night sky would be so much prettier.
5. Exiting Equilibrium: The presence of the moon plays an integral part in alot of things on planet earth, one of which includes our weather and climate. The earth rotates about its axis possibly because of the moon and if you remove that from the equation then get ready for a slew of weather problems which will bring about the mother of all climate changes which could maybe even turn Africa to the new Antarctica.
The bottom line is that, apart from a better view of the stars, nothing good can come from the moon disappearing.
Great answer, you nailed it
I'm not even going to try to better this one