How and why does atomic radius vary in the periodic table?

Atomic radius increases down a group in the periodic table. This can be explained by a phenomenon called electron shielding. As the number of electron shells is increased down the group, the attraction between the nuclear protons and the electrons is weakened and thus the atom is held together less tightly.

Across a period, the atomic radius decreases. This is because the number of protons increases as we proceed from the left to the right of the periodic table. (Look at the atomic numbers, Li=3 Be=4 B=5 etc.) More protons means more pull of electrons towards the nucleus, resulting in smaller atomic radii.