Learn Turkish: Kadın balkonunda kitap okuyor. - The woman is reading a book on her balcony.

This is exactly what this lesson is about! The word breaks down like this: balkon (balcony) + u (her) + n (buffer letter) + da (on/at). Whenever you add a case ending (like the locative -da) right after a third-person possessive suffix (like -u, -ı, -su, etc.), you must use n as a buffer letter to connect them.

How do we know balkonunda means "on her balcony" and not "on his balcony"?

Out of context, balkonunda could mean "on his balcony", "on her balcony", or "on its balcony". Here, it makes the most sense to translate it as "her balcony" because the subject of the sentence doing the reading is kadın (the woman).

The translation says "a book", so shouldn't there be a bir before kitap?

You could say bir kitap okuyor, which emphasizes that she is reading exactly one book. However, in Turkish, it is very common to drop bir for everyday routines and general objects. Kitap okuyor acts almost like a single idea: "She is book-reading."

Why doesn't kitap have an ending to show it's the object, like kitabı?