An eclipse can only happen when the Moon is at new or full phase while the line of nodes is pointing towards the Sun.  Due to a slight precession of the Moon's orbit, the line of nodes can point towards the Sun three times during one year. The geometry of solar eclipses is such that it is possible for the Moon to make a complete synodic orbit before the line of nodes moves too far out of alignment.  This means there can be two solar eclipses at each nodal alignment, except for the third one where the end of the year comes before there is time for a second eclipse.  Thus, there is potential for up to five solar eclipses per year!  Considering only umbral lunar eclipses, the geometry is such that it is possible for only one lunar eclipse during a complete synodic orbit of the Moon.  This means there can only be three potential lunar eclipses per year.