New Moon (Ammavasai) happens when the Moon comes roughly in between the Sun and the Earth. At this position, we get to see the face of the Moon that is opposite the Sun and hence dark. If this alignment happens perfectly, we get the Solar Eclipse (which is Moon blocking the Sun - partiallly or totally). In this phase, the Moon and the Sun rise and set at about the same time

Waxing Crescent Moon - As the Moon moves around the Earth, we get to see more and more of the bright half. At first we get to see a slice of it (thin crescent). This grows day by day.

First Quarter Moon - About a week after the New Moon, when the Moon has completed about a quarter of its turn around the Earth, we can see half of the brighter side of the Moon.

Waxing Gibbous Moon - During the following week, we keep seeing more and more of the brighter side of the Moon. It is now called the Waxing Gibbous Moon(gibbous means "humped")

Full Moon (Pournami)- About Two weeks after the New Moon, the Moon is now halfway through its revolution, and now the bright half of the Moon is totally visible, This is Full Moon. At this position, the Moon rises roughly at the same time the Sun sets, and it roughly sets when the Sun rises. If the Moon happens to align totally or partially with the Earth and Sun, we get a Lunar Eclipse

Waning Gibbous Moon - From now on, until it becomes New again, the bright side of the Moon keeps decreasing

Last Quarter Moon - About three weeks after the New Moon, we again can see half of the bright side of the Moon.

Waning Crescent Moon - Finally, after about four weeks, the Moon reduces to a thin size (Crescent) to become a New Moon again