Questions & Answers about Daj mi taj šešir.
Why is the word for "that" taj here, instead of onaj?
In Croatian, demonstratives depend on distance. Ovaj is for something near the speaker (this). Taj is for something near the listener (that). Onaj is for something far from both people (that over there). Since you are asking the person to give you the hat, it is near them, so we use taj.
Why doesn't taj šešir change its ending in the accusative case?
Taj šešir is the direct object of the verb, so it is in the accusative case. However, šešir is an inanimate (non-living) masculine noun. In Croatian, inanimate masculine nouns and their attached pronouns look exactly the same in the accusative case as they do in the nominative (dictionary) case.
Why do we use mi instead of meni?
Mi is the short, unstressed version of the dative pronoun meni (to me). In everyday speech, you use the short form unless you want to put special emphasis on "to ME" (and not someone else). Because it is a short word (enclitic), mi strictly takes the second position in the sentence, right after Daj.
What form of the verb is daj?
Daj is the imperative (command) form of the verb dati (to give). This specific form is used when you are giving an instruction to a single person you address informally (ti).