mam kurtki - means I have jacekts
nie mam kurtki - i don't have a jacket
Verbs mieć - (ja mam) and nosić (ja noszę) require a noun in Accusative form, but negation requires Genitive
By the way, the difference (in English) between jacket and coat is as follows: a coat (or overcoat) is a long, usually below the knee, heavyish garment, same also applies to a raincoat, or mackintosh; a jacket is a short garment in any fabric which traditionally stops either at the hip or top of the thigh. But as fashions change, descriptions get garbled and there are now 'long' jackets, car coats, bombers...the list is endless. But they are all relatively short, compared to an overcoat !