This guide has been compiled based on pure experience. Though I am unsure what exactly each personality makes it want to do, I am recording here the events I have seen each personality do most often.
Cry Baby: Basically this means the chao will tend to cry a lot. The personality may often change to this due to spoiling I believe. Though this is hard to confirm.
Short Temper: This chao is usually how the same says, on a short temper. It will easy get mad, and it's usual action is to stop around much like the chao "Wacky" does from Karate mode (Expert last level, Super first level).
Energetic: This chao doesn't get much sleep, it's usually to busy running around and basically more likely than not on the go. This is the chao you will see swimming for awhile or running around playing with a ball the most.
Big Eater: This chao isn't as picky about food. Infact, hardly will you ever find a time it isn't willing to eat. An example of this could be when you are trying to feed a chao and it just keeps throwing the fruit back! Well this chao shouldn't have a problem with eating at all, as it's a "big eater". Naive: This chao tends to cry a lot. Though naive as a definition is inexperienced and uninstructed. So I'm guessing by this that this chao must often not know what to do, and just roams around freely.
Carefree: As the same says, this chao is pure care free. It has realy not much to worry about, not that chao should except when they are going to eat. I am unsure of what a carefree chao may actually do. Curious: Forunately, curiosity doesn't kill the chao. This chao is often exploring and finding new things to do.
Nothing: Sometimes when you visit the chao, you may see that your chao has no personality trait. It is simply undefined. I believe this means that your chao is a mix of the above. So, it does have personality, just.. they happen to have the multiple personality disorder. Ofcourse though that is a joke, I have seen this type of chao doing a lot of the above.