Learn Polish: Pies biega po podwórku. - The dog is running around the yard.

You are right that po often means "after" when talking about time, like in the phrase po pracy (after work). However, when we talk about movement in a physical space, po translates to "around," "all over," or "along" a surface. In this sentence, the dog is running all over the surface of the yard.

Which grammatical case is podwórku, and why?

It is in the Locative case. When po is used to describe moving "around" a space or an event happening "after" a certain time, it requires the noun to be in the Locative case. The basic dictionary form of the word here is podwórko (yard).

Why does the Locative form of podwórko end in -u instead of -e?

Most neuter nouns take an -e or -u ending in the Locative case. Nouns that end in -ko, -go, or -cho (like podwórko or jabłko) always take the -u ending. This is a spelling rule that helps avoid awkward pronunciation changes that would happen if an "e" was added after a "k".

Could I use the verb biegnie instead of biega?