Grief means intense sorrow. You feel grief if something terribly sad happens, like if your dog dies or if your childhood sweetheart breaks up with you.
The word grief comes from the Latin word gravare, which means to make heavy. Gravare itself comes from the Latin word gravis, which means weighty. So think of grief as a heavy, oppressive sadness. We associate it most often with mourning a loved one's death, but it can follow any kind of loss.