Learn Polish: Nie mam czasu wolnego.

The verb mieć (to have) normally takes the accusative case, so "I have time" is Mam czas. However, in Polish, whenever a direct object is used in a negative sentence, it switches to the genitive case. Because of Nie mam (I don't have), czas changes to its genitive form, czasu.

Why is the adjective wolnego placed after the noun czasu? Shouldn't it be wolnego czasu?

In Polish, adjectives usually come before the noun if they simply describe a trait (like "a nice day"). But if the adjective establishes a set phrase, a classification, or a specific category, it often goes after the noun. Czas wolny (free time) is considered a fixed category, so the adjective comes second.

Why does the adjective wolny change to wolnego here?

Adjectives always have to copy the case, gender, and number of the noun they describe. Since czasu is in the masculine genitive case, the adjective must also take the masculine genitive ending, which is -ego. This turns the basic dictionary form wolny (free) into wolnego.