Like most things, Chao can die... A Chao's life usually lasts about 7 - 8 hours on the Dreamcast.
On the Gamecube the Chao has an age. When the Chao reaches 5 years old* it will reach the end of its life,
and it will cocoon and die after that. But, if your Chao liked you, it will reincarnate and leave an Egg.
If it didn't, then sadly your Chao will not reincarnate and will be gone forever.
*A Chao Year on the Gamecube lasts around three real life hours.
If you care for your Chao extra well, it's years will take longer.
Likewise if your Chao is unhappy and get hit alot, it will take shorter to age and in turn, die quicker.
Also note, that some studies show that when a Chao breeds with another Chao, its lifespan decreases.
The age of your Chao is shown in the Medical Chart with the Chao Doctor (in the Kindergarten)
When a Chao reincarnates on the Dreamcast, it keeps its name, race badges and personality,
but it loses all its skills, so it is just like a newborn Chao! (With 0 stats).
On the Gamecube version, it's just like the Dreamcast version, except it also keeps its grades.
However its stats go down to level one, and only keeps 10% of what they were before.
You may want to look at the stats part in the Gamecube section for more details on this.
A few more version differences. The picture below is from the Dreamcast version.
All Chao that die will look like this, and then you will have to wait for the cocoon to dissapear before you know if it has left an egg or not.
On the Gamecube, the cocoon will be pink coloured if it is going to reincarnate, and grey coloured if it is about to dissapear forever.
NOTE: (Dreamcast Version) If your Chao reincarnates, Do not take it to the Chao Doctor in the Kindergarten!
Doing so could result in the game glitching and it will freeze, and that Chao has the possibility of being deleted!
This seems to be fixed in the Gamecube version.