If you've lost someone you love, you'll be desperately sad, or heartsick. Your beloved grandmother's death or your best friend moving far away can both make you feel heartsick.
Grief and loss are connected to the adjective heartsick. When someone's heartsick, they're not just bummed out — they are grieving and mournful. You could also describe this as being heartbroken or heavy-hearted. The heart is important here, because of the strong emotion associated with the word. Originally, heartsick literally meant "ill from heart disease."