Commiseration refers to feelings of sympathy for other people. If a friend's pet dies, you will probably want to express your commiseration.
If you are feeling commiseration for someone, they are probably going through difficult times. We feel commiseration when others suffer or feel pain and we understand that pain. We commiserate with them. The Latin root com- means "together with." Just as communication and community are words involving getting people together, commiseration is about feeling other people's pain.