What kind of gift is δώρο?
Is it like a birthday gift or an ability gift?
Both. Also, when talking about abilities, χάρισμα is commonly used as well.
(which is where the English charisma comes from).
I translated as: "Δεν έχω ένα δώρο" but it's incorrect to include "ένα"?
That would almost be like saying "I don't have one gift" in English. In this case, it's a sentence with a general meaning, so the indefinite article is ommited. ^.^