Learn Croatian: Pričvrstio sam vijenac na vrata. - I attached the wreath to the door.

The verb pričvrstiti (to attach) works just like dynamic placement verbs such as staviti (to put). Because you are moving the wreath onto the door, the preposition na is followed by the accusative case to show the destination.

What case is vrata here, and why does it end in -a?

vrata is in the accusative case to show the destination. Even though it ends in -a like a typical feminine singular noun, vrata (door) is actually a neuter plural noun that does not have a singular form. The accusative plural ending for neuter nouns is -a, which looks identical to its nominative form.

Is the direct object vijenac in the nominative or accusative case?

It is in the accusative case because it is the object receiving the action. However, because vijenac (wreath) is a masculine inanimate noun (a non-living thing), its accusative form looks exactly the same as its dictionary (nominative) form.

How would this sentence change if a woman were speaking?

The past tense verb pričvrstio ends in -o, which tells us the speaker is male. If a woman were speaking, she would change the past participle to its feminine form and say Pričvrstila sam vijenac na vrata.