What is the difference between periods and groups on the periodic table?

Periods are rows horizontally along the periodic table whereas groups are vertical columns down the periodic table.
Elements in the same group react similarly. E.g. the group 1 metals (found in the first column of the periodic table) react readily with air and water (this is the similarity) but these reactions become more vigorous / explosive as you go down the group (this is why they are similar but not the same). This is because elements in the same group have the same number of electrons in their outermost energy level (and electrons are responsible for how an element reacts)
Elements in the same period of the periodic table have the same number of inner most (also known as 'main') energy levels so have the same amount of electron shielding.