Learn German: Wir müssen den dicken Stoff waschen. - We have to wash the thick fabric.

Stoff is the direct object of the sentence, which puts it in the accusative case. Because Stoff is a masculine noun, its definite article changes from der to den. Whenever you have a definite article in the masculine accusative case, the adjective following it must take an -en ending, giving us den dicken Stoff.

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

The word den is your biggest clue! In the accusative case, only masculine nouns change their definite article. Feminine stays die, neuter stays das, and plural stays die. Since you see den, you know Stoff has to be a masculine noun (der Stoff).

Why is waschen pushed all the way to the end of the sentence?

This happens because of the modal verb müssen (must / have to). In German, when you use a modal verb, it takes the standard second position in the sentence, and the main action verb (waschen) gets pushed to the very end in its original infinitive form.

Is Stoff related to the English word "stuff"?

Yes, they are related! However, while Stoff can sometimes mean "material" or "stuff" in a broader sense, its most common everyday translation is "fabric" or "cloth", which is exactly what needs washing here.