When you hear a professional keyboard player, they often play different parts with each hand. The part you can usually sing is called the melody. The other part is a harmony and makes the music sound more full than if there was only a melody.
It often answers the melody, like two people having a conversation. The harmony part is usually simpler than the melody. In the following song, the right hand plays a melody and the left hand plays C and G as harmony notes.
In the following song, the left hand part uses only two notes which rock backwards and forwards. This gives the rhythm of the music a good feeling as well as harmonizing with the melody notes.