Dzięki is the "informal" form of "dziękuję". You generally only use it with people you know well.
I didn't even know the word, only in Slovak which is d'akujem, and i guessed, then its probably "dziekuje", and it was actually correct :D
Either dzięki or dziękuję (which is more of thank you)
I think, dziękuję means thanks resp. thank you where dzięki stands for thanks to resp. due to. In German danke bzw. dank.