Like most things, Chao can die... A Chao's life lasts for 5 years until it reaches the end of its life.
A Chao Year 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 (unconfirmed).
When reading this page, please keep in mind the version differences for PS3 and Xbox 360. On those versions, all cocoons are white, regardless of what is happening to the Chao.
If your Chao had a high enough happiness stat (greater than 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 your Chao starts going into a coccoon about to die, the cocoon will be pink coloured if it is going to reincarnate and grey coloured if it is about to disappear forever.
If your Chao reincarnates, it will keep its name, race badges, personality and 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 section for more details on this.
Saving a Chao
As previously mentioned, your Chao is in the process of dying forever when enters a grey 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 grey 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 cocoon is grey, then you should repeat the above process to further increase its happiness.