words - What's the difference between いいえ and English's "no"?

In "Japanese for Busy People I" page 2 (kana or romaji edition) it says that いいえ is 'virtually the same as "no"'.

It sounds like there are some differences, but they want to omit mentioning them for the time being. What differences are there?

Answer

The key to whether to reply with はい or いいえ is not whether the form of the question is positive or negative, but whether the question is asked in a way which expects a positive or a negative answer:

一緒に行きませんか？(expects a positive answer)
はい、是非一緒に行きましょう。

一緒に行かないんですか？(expects a negative answer)
はい、残念ながら行けません。

I guess this response could be regarded as cyclic, since the obvious next question would be when a question expects a positive or a negative answer. Usually positive forms expect positive answers and negative forms expect negative answers. The tricky ones are the simple negative-form+か (without んです or のです), which are often used for invitations, or to ask for agreement. An example of an invitation is given above. Below is an example of asking for agreement.

これ、おかしくありませんか？
はい、そうですね。おかしいですね。

Returning to いいえ, I would say that in practise, it's mostly used in the sense which gibbon describes. Using it as "No" sounds pretty strong. More often, the main verb/adjective is repeated in positive/negative as needed.