Resources: Google Maps

The Moon takes one month to orbit the Earth, and it takes one month to rotate on its axis. Because of this, the same side of the Moon always faces the Earth. Half of the Moon is always bathed in sunlight, just as half of the Earth always is (and as half of all objects in the Solar System are). However, at any given time, depending on where the Moon is in its orbital path around the Earth, we here on Earth may see all of the sunlit half, none of the sunlit half, or some portion of it. This day-of-the-month-dependent viewing of different fractions of the sunlit half of the Moon is what gives us the phases of the Moon. 

The diagram above shows the primary phases of the Moon. In this setup, the Sun is off to the right. The cartoon at the center shows the physical positioning of the Earth and Moon relative to each other (and the Sun) at the time of each Moon phase. The outer circle of images show what the Moon looks like during each phase for an observer on Earth in the Northern hemisphere. To start, when the Moon appears in the New Moon phase, the Moon is in between the Sun and the Earth. Sunlight falls on the Moon, but those on Earth cannot see any of that light. The Moon orbits counterclockwise around the Earth as seen from space looking down on the north pole of the Sun, Earth, and Moon. As it does so, some of the lit half of the Moon begins to be able to be seen on Earth, appearing on the right hand side. This phase is called Waxing Crescent (waxing meaning 'growing larger').

The Moon then continues in its orbit until it has traveled one quarter of the way around the Earth. Half of the lit half of the Moon can be seen from Earth; the entire right half of the Moon in the sky is visible. We call this phase First Quarter. As the Moon continues to orbit around the Earth, more than half of the view of the Moon that can be seen becomes lit. We call this phase the Waxing Gibbous. Then, the entire Moon as seen in the sky is lit up. This is the Full Moon. The entire face of the Moon in the sky is lit because the Moon is behind the Earth relative to the Sun. The Sun lights up half of the Earth and half of the Moon, and we can see all of that lit half of the Moon here on Earth during the Full Moon. As the Moon continues in its orbit around the Earth, the lit portion of the Moon begins to disappear. Darkness begins to appear on the right hand side of the Moon. We call this phase Waning Gibbous (waning meaning 'growing smaller'). Next, when the Moon has traveled through three-quarters of its orbit around the Earth, half of the lit half of the Moon can be seen; this time the left side of the Moon as seen in the sky is lit. This phase is called Third Quarter (though it is sometimes also called Last Quarter). As the amount of the lit half of the Moon that can be seen decreases further, the Moon enters the Waning Crescent phase. Finally, the Moon returns to the New Moon phase, with the Moon in between the Sun and the Earth in space. 

To recap, the phases of the Moon are: Waxing Crescent, First Quarter, Waxing Gibbous (during these three phases, the amount of the lit half of the Moon seen is increasing each night), followed by the Full Moon, followed by Waning Gibbous, Third Quarter, and Waning Crescent (during these three phases, the amount of the lit half of the Moon seen is decreasing each night), followed by the New Moon. The cycle then continues with Waxing Crescent, etc. again. 

It is possible to understand the time at which each phase of the Moon will rise, set, and be at its highest in the sky (on the meridian) without having to memorize those details; this is done by thinking about the positions of the Moon, Earth, and Sun relative to one another during different times of the month as well as thinking about what sunset, sunrise, and having something high in the sky physically mean in terms of the Earth's orientation in space. However, charts of rise, set, and meridian crossings for each phase of the Moon can be readily accessed when needed:

The lunar month (the synodic month, the time it takes to complete a full lunar cycle relative to the Sun (from New Moon to New Moon or Full Moon to Full Moon, etc.)) is around 29.5 days, making it slightly longer than what we call a sidereal month, the time it takes for the Moon to complete the 360^o journey around the Earth (complete an orbit around the Earth relative to the background stars). This sidereal month is about 27.3 days long. This difference is because as the Moon orbits the Earth, so too the Earth orbits the Sun. The lunar cycle always has to do some catch-up and travel a bit more than 360^o to get back to the phase it started at.