"si" is only used to answer a negative question or assertion and contradict it: (Contredit un énoncé négatif - see the third tab [SI³, adv.])

Tu ne manges pas?
Si, j'arrive.

Tu ne fais jamais la vaisselle.
Si, je l'ai faite hier.

"oui" is used to answer a positive question:

Tu veux du pain?
Oui.

"si" is only used to answer a negative question or assertion and contradict it: (Contredit un énoncé négatif - see the third tab [SI³, adv.])

Tu ne manges pas? (Are you not eating?)
Si, j'arrive. (On the contrary, I'll be right there)

Contrast with the negative answer:

Non, je n'ai pas faim. (No, I'm not hungry)

Another example, not on a question.

Tu ne fais jamais la vaisselle! (You never do the dishes!)
Si, je l'ai faite hier. (On the contrary, I did them yesterday)

vs.

Non, et alors. (Indeed I don't, so what?)

"oui" is used to answer a positive question:

Tu veux du pain? (Do you want some bread?)
Oui, merci. (Yes, thanks.)

vs.

Non, merci. (No thanks.)

Also, note that en effet can be used to agree with the other person, regardless of whether he used a positive or negative sentence/question:

En effet, je n'ai pas faim. (Indeed, I'm not hungry)
En effet, je ne fais jamais la vaisselle. (Indeed, I never do the dishes.)
En effet, je veux du pain. (Indeed, I want bread — although that one is weird, unless you got some golden tattoo on your forehead.)