So you use "wiem" if you are aware of something, if you have knowledge of something. For example "I know a lot about biology" - "Wiem dużo o biologii", "I know it's true" - "Wiem, że to prawda" And you use "znam" if you know somebody, something. For example "I know Kate very well" - "Znam Kate bardzo dobrze", "I know the lyrics of this song" - "Znam słowa tej piosenki". I hope you understand now :)

I'm not an expert in Polish, so let's see if others answer better. It's a bit hard to explain translating to English since in English both mean to know. However there is a slight difference. Other languages do have this difference: in Spanish saber/conocer in German wissen/kennen, in Hungarian ismerni/tudni. I think in Russian you don't have the difference as you would use "знаю" in both cases. znam will go more with objects, things, people you know. --> Я тебя знаю (using in general the accusative case, or direct object of the action) wiem will be more about the concept of knowing something --Я знаю о чем ты говоришь (using a prepositive case) But let's see how other speakers can explain better

I just had a thought.. I agree with previous comments but to understand it easily I'd say: - if you use WIEM there is activity involved, for instance: 'I know how to make a cake' - wiem jak upiec ciasto 'I know you don't like me' - wiem, ze mnie nie lubisz - if you use ZNAM you refer more to the noun: 'I know this cake's recepie' - znam przepis na to ciast 'I know mr X and I don't like him' - znam pana X i go nie lubie Hope it makes sense ;)