"Waxing" means the lit portion is growing night to night; "waning" means it's shrinking. "Gibbous" just means more than half lit but not full.
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New Moon
The Moon sits between Earth and the Sun, so its lit side faces away from us. It's invisible, or nearly so, in the night sky, and marks the start of the cycle.
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Waxing Crescent
A thin sliver of light appears and grows a little bigger each night as the Moon moves away from the Sun's glare.
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First Quarter
Exactly half the face we see is lit. It's called "first quarter" because the Moon has completed one quarter of its trip around Earth, not because a quarter of it is lit.
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Waxing Gibbous
More than half lit and still growing night by night, closing in on full.
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Full Moon
The Moon is directly opposite the Sun from Earth, so the entire face is lit. The brightest and most familiar phase, the midpoint of the cycle.
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Waning Gibbous
Still more than half lit, but now shrinking each night on the way back down after full.
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Last Quarter
Half lit again, on the opposite side from First Quarter, three-quarters of the way through the cycle. Also called "Third Quarter."
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Waning Crescent
A thin, shrinking sliver just before the Moon disappears into the next New Moon and the cycle starts over.