Learn French: Il te donne l'ordonnance pour le médicament. - He is giving you the prescription for the medication.

In French, object pronouns like te (you / to you) are almost always placed directly before the conjugated verb. Even though English says "He is giving you", French structures it as Il te donne (literally: He to you gives).

Is te a direct or an indirect object here?

It is an indirect object. The verb donner (to give) usually works by giving something to someone. In this sentence, the direct object (the "something") is l'ordonnance, and the indirect object (the "someone") is te. Because you are giving it to someone, te acts as an indirect object pronoun.

Does ordonnance mean "ordinance" or a law?

It can in legal contexts, but at the doctor's office, une ordonnance is the standard word for "a prescription". You can think of it as the doctor "ordering" a specific treatment for you.

Why use médicament instead of "médecine" for medication?

Le médicament refers to the actual physical pill, syrup, or drug you take. La médecine refers to the science or profession of medicine, so you wouldn't use it to talk about the drugs a doctor prescribes.