Learn Polish: Jest mi zimno. Zamknij okno. - I am cold. Close the window.

In Polish, to express physical sensations like being cold, hot, or feeling sick, you use the verb jest (it is) + a dative pronoun like mi (to me) + an adverb like zimno. If you translate "I am cold" word-for-word as jestem zimny (or jestem zimna for a female), it actually means your skin is physically cold to the touch, or that you are an emotionally cold person!

Why use mi here instead of mnie?

mi is the short, unstressed form of the dative pronoun for "me", while mnie is the long form. You use the short form mi in normal sentences unless you are trying to heavily emphasize the "me" part (for example, "It is cold to me, but not to him") or if the pronoun is the very first word in the sentence.

What verb form is zamknij?

zamknij is the imperative (command) form for the informal "you" (singular). It comes from the perfective verb zamknąć (to close). You use the imperative when you are directly telling or asking someone to do an action.

okno is the direct object being closed. Why doesn't its ending change?

okno (window) is indeed the direct object, meaning it is in the accusative case here. However, because okno is a neuter noun (it ends in -o), its accusative form is exactly the same as its basic nominative (dictionary) form. It does not change.