Her yes is no, no is a maybe
Her language is so hard to learn
- Ace of Base, "Tokyo Girl"
There are many ways to say "no" in Japanese, and very few of them are direct. As Dave Barry managed to figure out, "maybe" and "possibly" both mean "no" in Nippon, and if someone says that an idea is "interesting," they really mean that it's stupid. But here are the most basic ways to express the concept:

Iie. = "NO."

This is the textbook translation of our English word "no." It is very rarely used in conversation... I think I can count the times I've heard it on my fingers and toes. Most people find iie to be too blunt for public use: the one place you will find it is on questionnaires, when it is used as the opposite of hai.

According to gn0sis, "iie is best used for polite denials: 'Hashi wa o-joozu desu ne!' ... 'Iie, heta desu!'."

Iya! (in an exasperated voice) = "HELL no!"
This is the sort of word you can expect to hear when you attempt to kiss someone who isn't interested, or when someone's bag falls on the tracks as the train approaches. Dictionaries gloss it as meaning "disagreeable" or "detestable," and the word can be used as an adjective with the particle na, as in iya na yatsu "creep."

In my experience, though, iya is just as common as an interjection, if not more so. It's slightly vulgar, so you'll probably only hear it among friends.

In the Kansai dialect, akan means the same thing.

Chigau! = "Wrong!"
I love chigau, mostly because dictionaries will tell you it means "differ," and won't be lying. The Japanese language makes little distinction between "different" and "wrong," which might explain part of the famed Japanese "herd mentality."

Chigau is what you're most likely to hear if someone is disagreeing with an opinion. You can use it in all the myriad ways "wrong" is used in English, but chigau is much more common in Japanese than "wrong" is in our language. Chigaimasu is the polite form: Kansai people contract the word to chau.

Chotto... = "Uhm..."
This word literally means "a little." When accompanied by lots of sucking air through the teeth and a trailing off at the end, it means "I would say no, but I don't want to offend you."

Chotto is a good word to use when turning down food, talking to your boss or teacher, or explaining yourself to the friendly traffic cop.

Dame = "It is forbidden!"
This is not pronounced like the English word "dame," but rather as "dah-may." It is what parents say to their children: if little Taro asks his mother "Senbei tabete ii?" ("Can I have a senbei?"), "dame" would be the expected response of the mother, at least right before suppertime. Teachers and other authority figures are fond of using the phrase as well: Kansai people will say akan instead. (Akan, in case you haven't gathered already, is a very useful word in Osaka.)

Incidentally, the phrase "Just Say No" is rendered in Japanese as Dame, zettai, which means "forbidden, absolutely."

Nononononono!
This is what Japanese people will say to you if you look like a gaijin.