Learn Hardcore Turkish: Kapı kapandı.

Turkish doesn’t have separate words for a or the. Definiteness (whether you mean “a door” or “the door”) is understood from context, word order, or added words like bir (“a”) or demonstratives (bu “this”, o “that”). Here, Kapı kapandı simply means “The door closed.”

What case is kapı in?

It’s in the nominative (unmarked) case because kapı is the subject of the intransitive verb kapanmak. Subjects of intransitive verbs remain unmarked.

Why isn’t kapı marked with the accusative suffix -yı (as in kapıyı)?

The accusative suffix -ı/-i/-u/-ü marks direct objects of transitive verbs. Since kapanmak (“to close/become closed”) is intransitive, there is no object—and thus no accusative suffix.

What exactly does kapandı mean, and what tense is it?

Root verb: kapan- (“to close, to become closed”)
-DI: simple past tense (narrative past)
kapandı = “it closed” or “it became closed” in simple past.