First quarter is elongation ninety degrees, the Moon a quarter of the way round its cycle, with exactly half the visible disc lit. The name is a frequent source of confusion: it refers to the quarter of the cycle completed, not to a quarter of the disc being illuminated, and the same applies to last quarter.
The phases are produced by the changing angle between the Sun, the Earth and the Moon, and the quantity that defines them is elongation, the Moon's angular distance from the Sun as seen from the Earth. The complete round from one new moon to the next is the synodic month, about 29.53 days, and it is longer than the sidereal month of about 27.32 days in which the Moon returns to the same place among the stars, because the Earth has moved along its orbit in the meantime and the Moon must travel further to catch up.
At first quarter the Moon rises around midday and sets around midnight, so it is the evening Moon, high in the sky at sunset. The terminator, the line between light and dark, runs straight down the disc at this moment and is where lunar surface relief shows most sharply in a telescope, because the low sun angle there throws long shadows from craters and mountains.
The quarters are also where the Panchanga and the phase scheme most nearly agree: the seventh or eighth tithi falls close to first quarter, though the correspondence is approximate because a tithi is defined by elongation increments of twelve degrees and varies in length.
Sources AstronomyAndrew Fraknoi, David Morrison, and Sidney C. Wolff (OpenStax) with The Indian CalendarRobert Sewell and Sankara Balkrishna Dikshit