What determines the length of each period in the periodic table?

The periods (rows) correspond to the energy levels where electrons are found in atoms.

The first energy level (just an s sublevel) can hold up to two electrons. This period contains only two ; H and He.

The second energy level can hold up to 8 electrons (s and p sublevels) so this period has eight elements; Li, Be, B, C, N, O, F and Ne.

The third energy level can hold up to 18 electrons (s, p and d sublevels).

Here it gets a little tricky, the 3d orbital does not fill until after the 4s orbital. The third period still contains only 8 elements (Na - Ar) because the third period contains the 3s and 3p sublevels.

The fourth period (K-Kr) has 18 elements and contains the following sublevels; 4s, 3d and 4p.

The 5th period also has 18 elements (Rb - Xe) and contains the following sublevels; 5s, 4d and 5p.

The fourth energy level can hold 32 electrons (s,p,d and f sublevels) The 6th and 7th periods have 32 elements.

Period 6 (Cs-Rn) includes elements in the Lanthanide series (La-Yb) and includes the sublevels; 6s, 4f, 5d, 6p