Learn Polish: On nosi czarny kapelusz.

It actually is in the accusative case, which is used for direct objects. But kapelusz is a masculine inanimate noun (a non-living thing). For masculine inanimate nouns, the accusative case looks exactly the same as the basic dictionary form (nominative). So czarny kapelusz stays exactly as it is!

What is the infinitive of nosi and does it only mean to wear?

The infinitive is nosić. It primarily means to carry, but when we talk about clothes, it translates to wearing them. It implies a regular habit or a continuous state of wearing something, just like in English.

Could I say On ubiera czarny kapelusz instead?

Not quite! The verb nosić is used for the continuous state of wearing clothes. The verb ubierać means to dress someone (or oneself). If you want to say that he is actively putting the hat on his head right now, you would use the verb zakładać.

Can we drop the word On in this sentence?

Yes, you can! Because the verb nosi already tells us it is the third-person singular (he, she, or it), you can just say Nosi czarny kapelusz. Keeping On just adds a little bit of emphasis to who is wearing the hat.