I'm familiar with the periodic table / periodic system, but I wonder why it's called "periodic" since there seems not much periodic about (there seems to be little or no predictability of which elements are stable) and from physics I know Z-value of elements and that is not periodic either. So what is a "period" in this context?
I agree that the "periods" in the periodic table are not mathematically regular. I think the simplest definition may be that a period begins when a new s-subshell starts to fill. Recall that in the s, p, d, f subshells there are 2, 6, 10, and 14 electrons, respectively, so the periods have to get bigger over time and thus cannot be regular. As to the lack of predictability of stability, that may be true, but the periodic table is pretty good at predicting the chemistry of elements by collecting together similar elements into groups (the columns). This similarity is mostly expressed by the s and p subshells, which is where a lot of the reactivity occurs.
So what is a "period" in this context? To answer this directly, a period is a row of the table.
As trb456 indicates, the repeating periods reflect the filling-up of atomic orbitals. Because the valence electrons (those in the outer shell of an atom) determine many of the chemical properties, the periodic structure places atoms with similar valences in similar groups (vertical columns.)