The characteristic of something to move thru a certain number of cycles which exhibit repetition, in a certain period of time or space.
When applied to a regularly-repeating vibration, this number is called the vibration's frequency.
Many aspects of music exhibit periodicity:
sounds which we usually think of as being capable of carrying a melody generally have timbres composed of periodic spectra;
the notes which form harmony are usually conceived, to various degrees, as proportional frequencies; the proportions inherent in these harmonic systems, when graphed on lattice diagrams, can often be shown to fall within the boundaries of a periodicity-block;
many types of temperaments can be generated by an interval which forms a cycle (which may be open or closed). An example is the cycle of "5ths";
durations and accents are usually grouped into regularly-sized and -spaced units of meters, measures, phrases, etc.;
chord progressions are similarly delineated;
large chunks of a musical piece are generally repeated, forming easily-identifiable sections;
Other art forms may also exhibit periodicity, for example, Islamic tile art, architecture, Native American rugs, possibly cinema...but none seems to place as much emphasis on it as music. This is probably due to the importance of time as a dimension of musical perception.
Periodicity also plays a part in defining the finity of an intonational system. See periodicity-block.