I'm wondering how the Romans would have said "yes" as in "yes please" or "no" as in "no thank you". I don't know if they would have said it exactly like that, but what would they have said if they had to mean something like that?
In Classical Latin, there were no words exactly corresponding to "yes" and "no". Non and ne were negatives, but they needed to combine with other words (like "not" in English).
There were, however, particles which could be used to agree with something. Both ita and sīc meant "thus", and became words for "yes" in the Romance languages. So if someone asked if you were lost, for example, you could respond "Ita!" ("It is so!"). For a stronger "yes", add vērō ("truly").
"No" on its own was a bit more unwieldy to express. Minimē is "not at all", minimē vērō even stronger. Negō means "I deny it!", nōlī is "don't!".
Another way to respond to a yes/no question is to repeat the verb, in the positive for "yes" and in the negative for "no". So if someone asked "are you lost?" you could say "I am" (sum) or "I'm not" (non sum).
At least in a few cases, I have been taught that Ita or ita vero is correct for yes. I have seen it used when answering questions. As @Draconis pointed out, Ita means it is so, and vero means truly or something like that. However, ita seems to be a shorter, more casual form, not different in strength. Another tactic I have seen (rarely) is to just reply with a verb. In English, this would look like: Do you speak Latin? I speak it. A standard form of no (at least for beginners) is minime.