Learn Italian: Ti lavi il viso e usi l'asciugamano. - You wash your face and use the towel.

Lavi comes from the reflexive verb lavarsi (to wash oneself). The ti is a reflexive pronoun that matches the "you" form. In Italian, when you perform an action on yourself, you put the reflexive pronoun right before the verb. So ti lavi literally means "you wash yourself".

The translation says "your face", but doesn't il viso just mean "the face"?

Yes, il viso literally means "the face". In Italian, when talking about body parts or clothing, you usually use the definite article (il, la, etc.) instead of a possessive word like tuo. Because you already used the reflexive pronoun ti earlier in the sentence, it is perfectly clear whose face is being washed!

Why do both lavi and usi end in -i, and where is the word for "you"?

Both verbs end in -i because they are conjugated in the present tense for tu (you, singular). Since that -i ending already tells us exactly who is doing the action, Italian usually drops the subject word tu. You could say tu ti lavi, but leaving it out is much more natural.

Why is it l'asciugamano instead of il asciugamano?