The First Quarter is the phase at which half of the Moon's disc appears lit, the right half as seen from the northern hemisphere, and the illumination is increasing toward full. Its name comes not from the shape but from the Moon having completed the first quarter of its monthly circuit around the Earth; the Sun, Earth, and Moon stand at a right angle. The first-quarter Moon rises around midday and sets around midnight, riding high in the evening sky. Traditionally it is read as a phase of decision, effort, and the overcoming of obstacles, the point where growth meets resistance. These are the astronomical facts of the phase together with the meanings traditions attach to it; the half disc is simply our side-on view of the sunlit hemisphere of the Moon.
The right angle the Sun, Earth and Moon form at first quarter is the same geometry astrologers call a square when it occurs between two planets, and the coincidence is not accidental: the classical aspects were originally defined by dividing the circle into simple fractions, a quarter, a third, a sixth, and the Moon's own quarter phases are the same division applied to the single fastest-moving body in the sky rather than to two planets at once. Traditionally minded astrologers sometimes read the first-quarter Moon as carrying something of the square's characteristic tension for exactly this reason, obstacle met and pushed through rather than avoided, treating the lunar cycle as a miniature, monthly rehearsal of the same hard-aspect dynamic that unfolds over much longer periods between the outer planets.
Sources AstronomyAndrew Fraknoi, David Morrison, and Sidney C. Wolff (OpenStax) with A History of Horoscopic AstrologyJames Herschel Holden