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?