Learn Polish: Dyrektor widzi, że jesteś zajęty. - The director sees that you are busy.

Yes! In Polish, you must always put a comma before the conjunction że (that) when it connects two parts of a sentence. It is a strict punctuation rule, unlike in English where commas before "that" are often left out.

Where is the word for "you" in the second half of the sentence?

In Polish, you do not need to use the pronoun ty (you) because the verb jesteś (you are) already tells us exactly who is doing the action. It is very common to drop pronouns like this unless you want to emphasize who you are talking about.

What if the director is talking to a female employee? Does zajęty change?

Yes. The word zajęty is the masculine form of the adjective "busy". If the director is seeing that a woman is busy, the sentence would end with the feminine form: Dyrektor widzi, że jesteś zajęta.

Does dyrektor mean a movie director, like in English?

It can, but in everyday Polish, dyrektor is most commonly used for the principal of a school, a company manager, or a CEO. A movie director is more specifically called a reżyser.