Death and Reincarnation
Like most things, Chao can die... A Chao's life usually lasts about 7 - 8 hours, and it will cocoon and die after that. However, if your Chao had a high enough happiness stat (over 30), it will reincarnate and leave an Egg. If it didn't, then sadly your Chao will not reincarnate and will be gone forever.
When going to the Chao Doctor, a Chao that has been alive for 0-5 hours will result in the statement "This Chao is growing". 6+ hours will show "This Chao has had a long life". If you care for your Chao extra well, it will live longer. Likewise if your Chao is unhappy and get hit alot, it will die quicker. Also note, that some studies show that when a Chao breeds with another Chao, its lifespan decreases.
If your Chao had a high enough happiness stat, it will reincarnate and leave an egg. If it didn't, then sadly your Chao will just disappear instead of turning into an egg again and will be gone forever. When your Chao starts going into a coccoon about to die, the Chao will shake its head while forming the cocoon if the happiness level is too low to reincarnate.
When a Chao reincarnates, it keeps its name, race badges and personality, but it loses all of its skills and will be just like a newborn Chao! (With 0 stats). The Gamecube version works differently to this. Please check that section for details.
If your Chao reincarnates, if you take it to the Chao Doctor in the Kindergarten could result in the game glitching and freezing. This can even result in that Chao being deleted! This is fixed in the GameCube version of the game.
Saving a Chao
As previously mentioned, your Chao is in the process of dying forever if it is shaking it's head while forming the cocoon. It may be possible to save the Chao if you act quickly. The first step is to leave the Chao Garden before it dies.
Then, re-enter the Chao Garden and quickly pick up the Chao before it recreates its cocoon. This can be difficult to do, and it might take several attempts. An easier method is to keep re-entering the Chao Garden until you find that the Chao has been randomly placed in a water location, as Chao cannot die while they are swimming.
Once you have managed to pick up the dying Chao, you can attempt to increase its happiness so that it reincarnates. To do this, put it back down on the ground and immediately feed it or pet it repeatedly. Please be aware that if you leave the Chao alone (just a few seconds is enough), it will form another cocoon and begin to die. If this happens and the Chao is shaking it's head, then you should repeat the above process to further increase its happiness.