The Moon - and to a lesser extent, the Sun - causes the tides on Earth.
At the time of the full Moon and of the new Moon, the Sun and Moon are aligned, so they pull together on the Earth. At the time of a first or third quarter Moon, the gravitational pull of the Sun and Moon are at right angles to each other.
When they pull together, we get spring tides
which are higher than usual. When they pull at right angles to each other, we get neap tides
which are lower than usual.
This animation shows how tides vary with Moon phase
. Watch the tidal bulges of both the Sun and the Moon.
On March 11, 2011 Japan suffered a devastating earthquake and tsunami. Some claimed that it was caused by a "supermoon" which would occur on the 19th. In fact, on March 11, the Moon was just coming into its first-quarter phase, i.e., when its gravitational influence on Earth is at its lowest. Unsurprisingly, no correlation has been found between perigee full moons - aka supermoons - and major earthquakes.