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!

Yes it's Black Saturday, and my yellow P1 Tamagotchi Kuchipatchi decided it was time to move on to the next life.  At age 11, it is pretty old for a Kuchipatchi, since back then, this is one of the unhealthy characters.  It's the best unhealthy character you can get though.

For those who've never experienced this 1997 Tamagotchi, there is no breeding function in the pet so there are no actual generations found in this toy.  It's either they move on to the afterlife because of old age, or die due to terrible care.

So your Tamagotchi turning into an angel is actually a good thing.  It means that you've taken care of your pet to its fullest.  Seeing a ghost with a grave on the other hand means that your tamagotchi died before its time signifying that you didn't take care of it properly.

When its near death, this is what happens:
1. Your tamagotchi gets sick (a skull icon appears). Several short beeps are heard.  If you try to press any buttons, it wouldn't respond.
2. The beeps get a bit longer and longer from each interval, just like the sound of a heart monitor in a hospital.  It will eventually end in a long beep.
3. Your tamagotchi lays/leaves an egg.  This will be your next pet.
4. Your tamagochi then turns into an angel with sparkly things beside it.  If you press the rightmost button, you will see the number of years he has lived in your care (as seen in the photo above).

Yes, he leaves behind an egg.  If you press the left and right buttons together, the egg will now be in your care and will soon hatch into another babytchi for you to take care of.