The First Quarter
Exactly half lit, and growing. Called a quarter because it is a quarter of the way through the cycle, which is the single most confusing piece of naming in astronomy.
Where to look: Rises around noon, is highest at sunset, and sets around midnight. The best-placed phase for anybody who wants to look at the Moon in the early evening without staying up.
The terminator — the line between lit and unlit — runs straight down the middle of the disc, and it is where every crater is at its most dramatic, because the Sun is rising there and the shadows are as long as they ever get. A first quarter moon through cheap binoculars beats a full moon through a good telescope.
Commonly muddled: Half the disc is lit, so people call it a half moon and then wonder why the calendar says quarter. Both are right about different things.