The ‘Saros Cycle’ is a particular approximation to an eclipse cycle. The ‘Exeligmos Cycle’ is an even better approximation.
Eclipses may occur when the Earth and Moon are on one line with the Sun, and the shadow of one body cast by the Sun falls on the other. Solar eclipses can potentially occur at every new (dark) Moon, and lunar eclipses at every full Moon. However, an eclipse does not happen at every new or full Moon, because the plane of the orbit of the Moon around the Earth is tilted with respect to the plane of the orbit of the Earth around the Sun (the ecliptic). An eclipse can only occur when the Moon is close to the plane of the orbit of the Earth, The points at which the Moon’s orbit crosses the ecliptic are known as nodes. The period of time it takes for the Moon to to complete an orbit from one node to the same node is known as the ‘Draconitic Month’. The Draconitic month is slightly shorter than the Synodic month. The main reason for this is that during the time that the Moon has completed an orbit around the Earth, the Earth (and Moon) have completed about 1/13th of their orbit around the Sun.
The ‘Saros Cycle’ is the most well known, and one of the best, eclipse cycles. It was discovered by ancient Babylonian astronomers several centuries BC and is based upon 223 synodic months equaling 242 draconitic months (to within 51 min). It is a period of about 6585⅓ days (approximately 18 years 11⅓ day) In the case of an eclipse of the Sun this ⅓ day means the region of visibility shifts west one third of the way around the world, and most places from which the first eclipse was visible do not see any of the second one. In the case of an eclipse of the Moon the next eclipse might still be visible from the same location as long as the Moon is above the horizon.
A more useful eclipse cycle is the triple Saros, or ‘Exeligmos Cycle’ which has the benefit that it has an almost integer number of days. So the next eclipse will be visible from a location close to the first one; in contrast to the Saros.