Full moon is opposition, elongation a hundred and eighty degrees, the Earth between the Sun and the Moon and the whole visible disc lit. Like the new moon it is an instant rather than a night, and the Moon appears full to the eye for a day either side of it.
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.
A lunar eclipse can only occur at full moon, and only when the opposition falls near a node of the Moon's orbit, which is the same condition that governs solar eclipses at conjunction. Because the two conditions are linked, eclipses come in seasons, and the interval at which the pattern repeats, the saros of about eighteen years and eleven days, was known to Babylonian astronomers and used for prediction.
The full moon is the anchor of a great many calendars and observances. Passover and Easter are tied to it through their computations, the Panchanga names the day Purnima and reckons the fifteenth tithi as ending at it, and in several Hindu calendrical conventions the month itself begins or ends at the full moon rather than at the new.
Sources AstronomyAndrew Fraknoi, David Morrison, and Sidney C. Wolff (OpenStax) with The Indian CalendarRobert Sewell and Sankara Balkrishna Dikshit, The Oxford Companion to the Year: An Exploration of Calendar Customs and Time-ReckoningBonnie Blackburn and Leofranc Holford-Strevens