To mourn is to grieve for someone who has died, especially a loved one. One of the hardest experiences of childhood is when you mourn the loss of a beloved pet.
You can also mourn for things that you've lost, not only pets and people who have died. If your local library branch is forced to close, you might mourn its loss, missing being able to walk there from your house. The Old English root word of mourn is murnan, which means not only to mourn, but also to be anxious. Related words include "mourner" and "mournful."