We're learning as the characters learn so there's no full and concrete truth of it yet - Sam learnt Dragon Glass worked against Walkers by stabbing one out of desperation and Jon *just* learnt that Valyrian steel works... this hasn't been covered in the book AFAIK because nobody's had a chance to hit one with a Valyrian blade yet.

Wights Vs White Walkers was explained in the first series when the two rangers' bodies are found, taken back to Castle Black and then re-animate as Wights and attack Jon and Lord Commander Mormont - they discover they're flammable by throwing a lamp at them after stabbing/chopping bits off doesn't have much of an effect.

Sam finds something or other in Maester Aemon's library that explains the WWs can reanimate the dead.