Learn French: Vous ne pouvez pas fumer dans le restaurant.

Pouvez is the vous (you, formal or plural) form of the verb pouvoir (can / to be able to). For example, je peux means "I can", but vous pouvez means "you can".

Why does pas go before fumer?

In French, the negative words ne and pas act like a sandwich around the conjugated verb. Here, pouvez is the conjugated verb, so it becomes ne pouvez pas. The second verb, fumer, stays outside the sandwich.

Why is it fumer and not fumez to match vous?

When two verbs appear together in French, generally only the first one is conjugated. The second verb stays in its dictionary form (the infinitive). Since pouvez is already conjugated for vous, fumer (to smoke) remains an infinitive.

Could I say au restaurant instead of dans le restaurant?

You could, but dans le restaurant emphasizes being physically inside the building. This makes perfect sense for a smoking rule, as smoking might be allowed outside on the patio, but not inside.