Confidence describes how your champion believes it will fare in a fight. This is independent of what you think, for you are not the one taking the scrapes and pains of the fight.
Confidence affects how well your commands are carried out.
Cofidence falls if a combatant is having little effect on its opponent. Normally, having attacks miss will cause increasing anger, but as the physical state of the combatant fails, he or she will question his or her own abilties as a fighter.
Another way to make confidence fall is to have a combatant face a physically intimidating opponent. A vast difference in size normally can do this. Another way to have this happen is for the combatant to face an opponent at an higher evolution level. This is the advantage I give to evolutions: they scare smaller, and normally less mature opponents. This strikes even truer if the opponent is an evolved version of the combatant, for while they will be more familar with their own species, and now without doubt an evolution must be somewhat stronger in the ways of battle.
A lose of confidence will have a slight negative effects on attacks. Combatants may hesitate to perform commands, and they may hold back a bit when doing said commands. If things are going really sour, the combatant may suffer total insecurity about their skill, and this may lead to sheer terror of their opponent. This does not bode well for battle: a combatant in this state may ignore commands due to a lack of faith itself or it may ignore commands to save its skin from an attack it believes with kill them.
Higher confidence is caused by just the opposite: if a battle is going very well or an evolution faces a smaller, "lesser" form, confidence may rise. This results in attacks being performed with a small increase in accuracy, power, and overall speed.
However, high confidence is good, but this can lead to cockiness. Rather than have high faith in themselves, they instead overestimatet heir own abilities and underestiamte their opponents. An arrogant combatant will perform attacks in a sloppy manner, believing they need not waste so much energy on such a "weak" opponent. Furthermore, they will let their guard done, allowing incoming attacks to do much more damage than normal. In extreme cases, an arrogant, cocky combatant may even out-right ignore commands that require exterting a lot of energy.
Confidence is fickle, however. A cocky combatant that suddenly does poorly can suddenly become very solemn and insecure, whereas a very scared combatant can perform particularlly well and gain a boost of confidence, and performs become cocky themselves.
Confidence will always be listed in the stats at the end of each round, if they are different than the average confidence of a battler. I have no real suggestions on how to battle around such mental conditions, but this is something you should prepare for when I ref your matches.