Learn Hardcore Dutch: De winkel is nog open.

Because winkel is a de-word. In Dutch, singular nouns take either de or het as their definite article, and winkel belongs to the de group.

So:

de winkel = the shop/store
de winkels = the shops/stores

There is not always a simple rule you can rely on, so it is best to learn nouns together with their article.

What does nog mean here?

Here nog means still.

So the sentence means that the shop/store is still open.

Be careful, though: nog can mean different things in different contexts, such as still, yet, another, or more. In this sentence, the natural meaning is still.

Why is the verb is in the middle?