The Chao personalities I have seen are:
(no personality) - no special characteristics
Cry Baby - Intelligence and Luck cannot be increased, Chao will be less content
Naive - Mates more willingly with any Chao, and gets double growth of Intelligence and Luck
Big Eater - always eats greedily
Picky Eater - often refuses food
Carefree - Chao will be more content
Curious - Double growth of Intelligence, but Luck cannot be increased
Energetic - performs behaviours more often
Short Temper - Double growth of Luck, but Intelligence cannot be increased
(no personality) Chao are the most common, and have no real problems; it's worth keeping them.
I tend to just get rid of Cry Babies when they're born; not only are they annoying to raise, but they are at a large disadvantage in racing because of their total lack of Int and Lck. You can change a Cry Baby's personality to Short Temper by beating it and making its life hell for several Chao years, but why go to the trouble? Chao life is cheap :P
Naive is the best personality if you raise your Chao to race. A Naive Chao will max his hidden stats twice as easily, and he also breeds more easily - saving you hassle and Heart Fruit when you want to pass on his genes.
Big Eaters max stamina very easily; Picky Eater, the inverse, can be annoying when you want to raise stamina. Luckily you can change a Picky Eater to a Big Eater by giving him nothing but basic garden fruits. You can also change a Big Eater to a Picky Eater by presenting him with a large variety of special fruits, but why would you want to?
It's harder than usual to make a Carefree Chao cry, and if there is an accident they'll be quicker to forgive you. Another nice personality, but, like the Big and Picky Eaters, they have no especial competition advantages.
Curious Chao are at a competition disadvantage, since Luck cannot be increased; given the choice, and despite Curious's Int growth, I would take (no personality) over Curious as (no personality) can max both stats instead of one.
Energetic Chao are one of the only two personalities that can become bullies. They have no competition advantages but are fun to have in the garden because their low boredom threshold means they're always up to something.
Short Temper is, predictably, the other bully personality, and this Chao becomes a bully most easily. While he's worth having just for this, his inability to increase Intelligence puts him in the same position as a Curious Chao where competition is concerned.
Your Chao's personality is said to be decided by the way you hatch him from the egg. In practice, I have not noticed any trends. I hatch all my Chao the same way, by rocking the egg for three to five seconds, and I get a cross-section of Chao personalities.
Chao personalities can change during evolution! I was raising an Energetic Chao to be a bully, and to my annoyance, when he evolved he had become Carefree. More study is needed in this area.