There comes a time when one will pass away.  Some due to old age; others due to other reasons.

In the picture above, the left one died earlier than its time.  I overslept one day and due to this, I was unable to attend to this tamagotchi.  I woke up hearing the "death beeps" and was shocked about the death of my first tamagotchi.  To think she was a mimitchi!

Up until this day, no tamagotchi has died under my care.  I guess there's a first time for everything.  And so, I get the bad owner death animation.  A grave is placed on the right side of the screen, with the tamagotchi's spirit floating beside it.  Sad.  Mimitchi still left an egg though for its successor.

The tamagotchi on the right is Bill.  He lived a very long life.  29 years to be exact (or 29 earth days -- almost a month!)  Taking care of him got very frustrating as almost every five minutes, all his hearts dropped down to empty and I had to feed and play with him until all are full again.  This is the tamagotchi's "old age" syndrome.  They get so needy that it's near annoyance.

And one day, he moved on.  As I was feeding him food, suddenly he got sick, the "death beeps" are heard, left an egg and turned into an angel.  Pressing the right button, it shows he lived up until the ripe old age of 29, a feat for Tamagotchi owners everywhere.  Great!

===EDIT===

Well, I stand corrected.  It just differs in region just like the person in our comments posted.  Thanks for the info!