Learn French: Mets les verres à vin sur la nappe délicate. - Put the wine glasses on the delicate tablecloth.

This is the difference between purpose and contents. Using à describes what the object is designed for, so verres à vin refers to the glasses themselves (wine glasses). If you said verres de vin, you would be talking about glasses that are currently filled with wine (glasses of wine).

Why is it à vin without an article, rather than au vin?

When à is used to show the purpose or function of an object, it is followed directly by the bare noun with no article. If we were describing a recipe's flavor or main ingredient, we would include the definite article, as in sauce au vin (wine sauce).

I thought we drop the -s at the end of tu commands. Why does Mets keep it?

You only drop the -s in the informal command form for regular verbs that end in -er (like mange or regarde). Since mettre is an -re verb, it keeps its standard present tense spelling for the command form (mets).

Can délicate go before the noun, as in la délicate nappe?

Yes, but placing it after the noun (la nappe délicate) is the standard, objective way to state that the fabric is fragile. Moving it before the noun is allowed in written French, but it makes the sentence sound more poetic and emphasizes your subjective appreciation of the tablecloth.