When is the Moon in its first quarter?

The phase cycle repeats roughly every 29.5 days, the interval between two new moons. First quarter marks the moment when the Moon has traveled a quarter of that monthly journey and sits halfway between new moon and full moon. Don't confuse that rhythm with the orbital one: the Moon completes one trip around Earth in about 27.3 days, but while it does, the Earth-Moon system also moves around the Sun, so the illumination cycle we see from the ground ends up slightly longer than the orbit.

At first quarter we see half of the lit face: half bright disk, half shadow. The Moon rises around midday, stands high in the sky at dusk, and sets at midnight, so the early part of the evening is the best time to observe it. The boundary between light and shadow, the terminator, crosses the disk right through the middle, and that's where surface relief shows the most contrast: long shadows pick out the rims of craters and mountain ranges, something that doesn't happen at full moon, when light arrives head-on and flattens the view.

First quarter and last quarter show the same illuminated fraction, but not the same half, and not at the same time. First quarter appears before full moon and dominates the evening sky; last quarter comes after and shows up in the early morning hours. The word "quarter" refers to the fraction of the cycle completed, not the portion of the disk visible, which is exactly half. Hence the apparent contradiction that something the eye reads as a half-moon is properly called a quarter.