Like most things, Chao can die... After living its life out as an adult Chao, it will eventually grow old and a cocoon will appear around it. This cocoon is identical to the one that appears around the Chao when it evolves from a child to an adult.
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 not reincarnate and will be gone forever.
When a Chao reincarnates, it will keep its name and its personality, but it loses all of its skills and will be just like a newborn Chao!