I've never had any pets, so haven't had to deal with them passing away. My friends have had pets that died, though, and it's surprising jus thow much it effects them.
If you've not had any pets, you won't understand. But they're members of a family, as much as any person is.
We had to have our cat, Tammy, put to sleep. She was very old, frail, and was getting pretty senile. It was really sad though.