it is very very complex. I'll try to give you a few examples. 1. when a statement is wrong. Let's say you wear a green sweater and someone says: Your sweater isn't green. In English, you'd say: yes it is! In german, you say: Doch! Or: you will come with us tonight, but someone asks: you are not coming? In english: yes I do come! (4 words!) In german you simply say: Doch! 2. You believe something truly, you think, the other is wrong. Der Test war so schwer! You: Der Test war DOCH einfach! (the test was so hard! , No, I think the test was easy) Ich werde 100 m in 8 Sekunden laufen! you: Das schaffst du DOCH nie! (I'll run 100 m in 8 seconds. You'll never do it!) Der Fernseher ist kaputt! You: Der funktioniert doch! (The Tv is broken! No, it works) There are other ways we use it, but it is hard to explain by writing it simply down, i'd need 100 words for it! Hope, I helped a bit. :)

I hope this is helpful for you: http://www.dict.cc/?s=doch

Doch, is the 'affirmative negation' to a sentence of negative conclusion if I'm right. Similar I guess to the word "si" in French. I think that is its major function. So, in English, although there is not a word of an identical function, one could say: 1. - there are no eggs left in the fridge! - not really, I think there's a couple of them left, have another look. - Es gibt keine Eier im Kühlschrank! - Doch, ich denke es gibt noch ein paar, guck noch ein mal. 2. - Your sister seems very sad, I heard she failed her exam. - Not really! she has indeed succeeded but it was really stressful for her. - Deine Schwester sieht traurig aus, hat sie doch ihre Prüfung nicht geschaft? - Stimmt doch gar nicht, sie hat ihre Prüfung doch bestätigt. Trotzdem war es stressig zu ihr. I hope my sentences are correct!

It's not that confusing. If someone says "no" and you want to say "yes", then you must say "doch". Don't say "ja"!