Compound tenses and moods are verb forms which are conjugated with two parts: a helping / auxiliary verb and a past participle, as in J'ai dansé. The word order can get a little complicated when additional grammatical structures like object pronouns and negation are introduced.
In English, two negatives are said to make a positive: that is, they cancel one another out, and this is grammatically unacceptable. In French, however, négation double is alive and well. Two negatives sometimes make a positive, while other times they combine to make the negation stronger or more specific.
French has three negative constructions that are reserved for formal (usually written) French like literature and historical accounts.
With articles, the de vs du, de la, des choice has to do with affirmative/negative and whether there's an adjective in front of the noun.
The rule is that to make a French verb negative, you need ne in front of the verb and pas after it. The reality of how the French speak says otherwise.
Inverting subjects and verbs is easy enough - vous voyez => voyez-vous, but where do object, adverbial, and reflexive pronouns go? And what about negation? Take a look at this lesson to learn about all the possibilities.