New and Full Moon
The most powerful tides occur during the new moon and full moon. This is because at these points the moon, earth and sun are lined up in a straight row, accentuating their gravitational pull. These are known as spring tides, though they occur during every season.
Half Moon
When the moon is only half lit, it means that it's perpendicular to the Earth. At this point the tide is lowest because the moon's gravitational pull is thrown out of line with the Earth and sun. These are known as neap tides.
Proxigean Spring Tide
The most powerful tides of all are the Proxigean spring tide, which occur roughly every 18 months. These are the result of the moon coming particularly close to the Earth during it's orbit, amplifying its gravitational force even more. The Proxigean spring tide only happens during the new moon, when the moon is between the Earth and the Sun.
Four Times Daily
Regardless of the phase of the moon, roughly four tides occur each day. These vary depending on where the moon is relative to the spin of the Earth. Since the Earth spins a full 360 degrees each day, each tide can be associated with a 90 degree turn. They shift from high tide to low tide and then back to high tide. These tides are most pronounced during the spring tides, while their effect is less noticeable during a neap.