Learn Farsi: قضیه چیست؟ - What is the matter?

While مشکل means "problem" and موضوع means "topic," قضیه translates more closely to "the case," "the situation," or "the story." Asking قضیه چیست؟ is like asking "What is the story here?" or "What exactly is going on?" rather than just asking about a specific problem.

چیست looks like written Farsi. How would people actually say this in everyday conversation?

You are spot on. چیست is the formal combination of چه (what) and است (is). In spoken Farsi, people blend these together into چیه (chieh), making the everyday spoken phrase قضیه چیه؟.

The English translation says "the matter", but there is no word for "the" in the Farsi sentence. Why is that?

Farsi does not have a direct equivalent for the definite article "the." Unless you add the indefinite suffix -ای to make it "a matter" (قضیه‌ای), a standalone noun like قضیه is naturally understood as definite ("the matter") based purely on the context of the question.

Would I use this sentence to ask a friend "What's the matter?" if they look sad or sick?

Not usually. قضیه چیست؟ is used to ask for the facts of a situation, similar to "What's the situation?" If you are checking on a friend's personal well-being or feelings, it is much more natural to ask چی شده؟ (What happened?) or مشکل چیه؟ (What is the problem?).