Last quarter is elongation two hundred and seventy degrees, three quarters of the way round the cycle, with half the disc lit and the lit half on the opposite side from first quarter. It is also called third quarter, and both names refer to the fraction of the cycle elapsed rather than to the fraction of the disc illuminated.
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 last quarter the Moon rises around midnight and sets around midday, so it is the morning Moon, visible in the daylight sky of the forenoon. The terminator again runs straight down the disc, and again this is when relief shows most sharply in a telescope, though the region under low sun is the opposite half of the surface from that seen at first quarter.
Because the quarters divide the lunation into four intervals of about seven and a third days, they are the most likely astronomical origin proposed for the seven-day week, although the week ran independently of the Moon in every historical system that used it and the connection is an inference rather than a documented derivation.
Sources AstronomyAndrew Fraknoi, David Morrison, and Sidney C. Wolff (OpenStax) with The Oxford Companion to the Year: An Exploration of Calendar Customs and Time-ReckoningBonnie Blackburn and Leofranc Holford-Strevens