Learn Polish: Tu nie można pić piwa. - One cannot drink beer here.

Because the sentence is negative. The basic word for beer is piwo, which you would use in a positive sentence like Piję piwo (I am drinking beer). But in Polish, whenever you negate a verb with nie, the direct object switches to the Genitive case. That is why piwo changes to piwa.

What exactly is można and how is it different from mogę?

Można is a special impersonal word meaning 'it is possible' or 'one is allowed.' It doesn't refer to a specific person. If you used mogę, it would mean 'I can.' Using nie można is the standard way to express general rules, like 'it is not allowed' or 'you cannot' do something.

Why does the verb pić stay in its basic dictionary form?

Pić means 'to drink' (the infinitive form). Just like in English, when you combine a helper word like można (one can / it is allowed) with an action, the main action verb stays in its basic form. You do not conjugate it.

Why doesn't the sentence say Nie można pić piwa tu?

Polish word order is flexible. You actually can put tu (here) at the end, but putting it at the beginning places emphasis on the location. It sounds more natural when stating a rule for a specific place, like 'Here, drinking beer is not allowed.'