Translation:My grandmother is cooking potatoes for lunch.
both are correct,
I have no idea why, but suposedly "ziemniak" was more used in south Poland (Austrian partition), and kartofel cenrtal-east (Russian partition) but then both russian and german word for potatoes are similar to kartofle
also prof Miodek says kartofel is much older word, http://www.gazetawroclawska.pl/artykul/387160,rzecz-o-jezyku-o-wyzszosci-kartofla-nad-ziemniakiem,id,t.html
That's more or less correct. The standard and most common word is ziemniaki, but most people may use kartofle interchangeably and everybody understands both words. There are other, regional words for potatoes as well, the most commonly known being pyry, used in Wielkopolska.