Learn Turkish: Kimi davet ediyorsun? - Whom are you inviting?

The word kim means who. We add the specific object ending (the accusative case) -i to make it kimi (whom) because the person being invited is receiving the action directly. You use kimi whenever you want to ask whom an action is being done to.

Why is the verb davet ediyorsun made of two words?

Many verbs in Turkish are formed by combining a noun with a helper verb. Davet means invitation, and the helper verb etmek means to do or to make. Together, they mean to invite.

The helper verb is etmek, so why does it change to ediyorsun here?

This is due to a rule called consonant mutation. When the root et- gets a suffix starting with a vowel, like the present continuous suffix -iyor, the hard t softens into a d to make it easier to say.

Where is the word for "you" in this sentence?

It is built right into the verb ending! The -sun at the end of ediyorsun means you (singular/informal). You could say Sen kimi davet ediyorsun?, but dropping the Sen is perfectly natural and more common in Turkish.