To grieve is to feel sorrow over something, especially people who have died.
The most common use of grieve has to do with death: usually there are grieving family members and grieving friends, feeling sadness over the loss of a loved one. You can also grieve over a pet who died, a job you didn't get, or a favorite pair of shoes that were ruined. Usually there's some kind of loss involved, but as long as you feel extremely sad about something that's happened, you're grieving.
