What is harmony?
It takes at least two voices to make harmony. One person sings the tune (the melody) and then harmony is created by adding another voice part, which is sung at the same time as the melody. Together, the two parts create a two part harmony. By adding 2 additional parts + the melody you have 3 part harmony and so on. Together all these parts create chords. The harmony parts should compliment the melody making the overall impression pleasing to the ear. That's the simple answer!
There are lots of different sorts of harmony singing out there. Some of the best harmony singing is done by acappella singers (unaccompanied singing), just the voices make up the chords. Barbershop singing is one of many genres of harmony singing they excel at 4-part harmony singing.