Elements in the same group are those that are in a single vertical line from top to bottom. They have the most in common between the two directions. They share the same number of electrons in their valence subshells. As a result, they will all have the same chemical reactivity. For example, all elements in group 1 have a single electron in the valence shells, so they easily give up that electron to become +1 cations.
Elements in the same period are those that are in a single horizontal line from left to right. They do not share the same type of chemical reactivity as seen in groups. They only real thing that these elements share in common is the same principal quantum number, or main energy shell. For example, period 5 elements have their electrons filling up the to the fifth energy shell and period 6 elements have electrons that go the sixth energy shell.