The main way the moon affects the Earth is the tides. The moon plays an important role in protecting the Earth from space rocks, such as meteorites. More subtle effects of the moon include minor effects on climate, the heat of the crust and the speed of the Earth's rotation.
The tidal effect of the moon only significantly affects the largest bodies of water, such as the major oceans. The gravity of the moon pulls the water in the oceans toward it. As the Earth rotates, the bulge shifts through the different regions of the globe. Tides appear on both sides of the Earth because of the pull of the sun.
The moon is an important buffer between the Earth and extraterrestrial bodies, which was more important during the initial formation when the Earth and moon were closer together. However, the moon is gradually shifting away from the Earth, so its protection is much less than in the past.
Minor effects are not very noticeable except when they are on a very large scale. For example, the length of a day on Earth is slightly longer now than it was in the time of the dinosaurs. The tidal forces also affect the crust, which contributes to the heat of the magma underneath. Tidal forces also keep ocean currents moving, which has a small affect on climate due to cool and warm water moving through the oceans.