Periodicity is best defined as:
the fact that elements increase in atomic mass in a regular way
the repeating nature of physical and chemical properties with atomic numbe
the regular growth of atomic size with atomic mass
the consistent decrease of nuclear charge with atomic mass
the repeating nature of atomic nuclear structure

Periodic table is designed in a way in which elements are divided in groups and periods. Periods are rows and groups are columns. The elements in groups share similar properties due to the same number of valence electrons they have. Going along each period, we'll always get back to the same group at some point. This is why we have periodic trends, the trends repeat gradually when we finally reach the same group we're analyzing.

Physical and chemical properties vary periodically. For example:

atomic radius decreases from left to right along the period;
ionization energy increases from left to right along the period;
electronegativity increases from left to right along the period;
metallic properties decrease from left to right along the period.