Learn Hardcore Turkish: Kupa kırıldı.

kırdı is the simple past of the transitive verb kırmak (“to break” something).
kırıldı adds the passive/anticausative suffix -ıl- to the root kır-, then the past tense -dı:
kır- + ‑ıl- + ‑dı → kırıldı
So kırıldı literally means “it was broken” or “it broke (by itself)”.

Why is kupa in the nominative case (no suffix) rather than the accusative?

In a passive or anticausative construction like kırıl-, the patient of the action becomes the grammatical subject. Subjects in Turkish appear in the nominative case (no ending). If this were active/transitive you’d say Kupayı kırdı (“He/she broke the cup”), where kupayı is accusative.

Does kırıldı imply someone broke it on purpose, or that it simply broke?

It’s ambiguous. As a passive it can mean “it was broken (by someone).” As an anticausative it can mean “it broke (by accident or on its own).” Context usually tells you which sense—if you care about the agent, you would specify it (e.g. Kupa Ali tarafından kırıldı).

How do you ask “Was the cup broken?” in Turkish?

Add the question particle -mı/-mi/-mu/-mü after the verb, adjusting for vowel harmony:
Kupa kırıldı mı?

What tense is kırıldı, and how does it differ from other past-tense forms?

It’s the simple past (di’li geçmiş zaman). You form it on the stem (here kırıl-) by adding -dı/-di/-du/-dü according to vowel harmony. This tense presents the event as a definite past fact.

Why doesn’t Turkish use an article like “the” in Kupa kırıldı?

Turkish has no articles (no direct equivalent of “a,” “an,” or “the”). Definiteness or indefiniteness is inferred from context. Here kupa could mean “the cup” or “a cup,” depending on the situation.

Can I specify which cup broke if there are several?

Yes. You can use a demonstrative or adjective:
O kupa kırıldı. – “That cup broke.”
Or add a possessive:
Mavi kupa kırıldı. – “The blue cup broke.”

How do you say “The cups broke” (plural)?

Pluralize kupa with -lar/-ler, then keep the same verb form:
Kupalar kırıldı.

What if I want to mention the person who broke the cup?

Use an active sentence with an explicit subject, or keep the passive and add tarafından:
Active: Ali kupayı kırdı. – “Ali broke the cup.”
Passive with agent: Kupa Ali tarafından kırıldı. – “The cup was broken by Ali.”