Learn German: Ich brauche den weichen Stoff.

This is adjective declension! Stoff is the direct object here, so it is in the accusative case. Because Stoff is a masculine noun, we use the article den. When an adjective comes directly after the masculine accusative article den, it must take an -en ending.

How can I tell the gender of Stoff just by looking at this sentence?

The article den gives it away. Stoff is the direct object (the thing you need), putting it in the accusative case. The only gender that uses den as its definite article in the accusative is masculine (der changes to den). Therefore, it must be der Stoff.

Why is a direct object needed after brauchen?

The verb brauchen (to need) requires a direct object because you always have to need something. In German, the direct object is always placed in the accusative case. Here, the fabric is the thing being needed, so the whole phrase den weichen Stoff is accusative.

Can I use Stoff to just mean generic "stuff" in German?