I am sure that you are aware of the three different genders we use for nouns in German, are you not? However, do not blame yourself in case you struggle with the articles! When working with non-native speakers I realised that this brings a lot of difficulties.
First things first: Every noun has a gender. For example: the man is masculine. The article for masculine Nominativ (you remember? We talked about this casus) is: DER. That is why it is: DER Mann. In contrast to that: the woman is feminine. The article for feminine Nominativ is: DIE. So, it is DIE Frau. The third gender is called Neutrum. The article for Neutrum Nominativ is: DAS. For example, "the animal" in Neutrum. So, it is "DAS Tier".
I am afraid you have to memorise the genders of the different nouns. There is no system behind it.