Why do some atoms form positive ions and others form negative ions?

It is energetically favorable for all atoms to have a complete outer electron shell. Loosely, the atoms on the left hand side of the periodic table only have a few extra electrons in their outer shell so it is energetically favorable for them to lose them. The atoms on the right hand side of the periodic table almost have enough electrons in their outer shell and so they have a tendency to gain them. Once electrons have left an electron shell, an atom will have a positive charge because it has more protons (positive charges) than electrons (negative charges). Similarly, an electron which has gained electrons to complete its outer shell will have a negative charge because it now has more electrons (negative charge) than protons (positive charge).