Learn German: Ich kaufe den dünnen Gürtel.

This is adjective declension in the accusative case. Because Gürtel (belt) is the direct object being bought, it is in the accusative. After the masculine accusative definite article den, adjectives always take an -en ending.

Why do we use den instead of der for "the"?

The belt is the direct object of the verb kaufen (to buy), so it takes the accusative case. Gürtel is a masculine noun, and the masculine definite article changes from der in the nominative (the subject) to den in the accusative.

Is Gürtel singular or plural in this sentence?

It is singular here. The noun Gürtel actually looks exactly the same in both its singular and plural forms! We know it is singular here because of the masculine article den. If you were buying multiple belts, the plural accusative article would be die, making the sentence: Ich kaufe die dünnen Gürtel.

How would the adjective ending change if I bought a feminine noun instead, like a bag (Tasche)?