You use "levantarse" when you talk about getting up, usually in the morning, but not exclusively. For example: Hoy me levanté a las 10 de la mañana. But also: Me levanté de mal humor de mi siesta (that's not in the morning). You can also use it as "stand up": Me levanté de la silla para que puedas sentarte. You use "madrugar" to talk about getting up of bed really early, like 5, 6, maybe 7 am (that's a bit subjective, right? :P). At least, that's what I consider early enough to have the right to say, the day before: "Me voy a dormir, ¡mañana tengo que madrugar!" Good luck with Spanish :)

And "madrugar" when you wake up really early, for example before 5 am to go to the airport: Hoy he madrugado para ir al aeropuerto.

Levantarse: Dejar la cama tras el descanso habitual (wake up) Madrugar: Despertar o LEVANTARSE temprano (rise early)

madrugar = levantarse temprano, por la mañana. Madrugo mucho, me levanto a las 5 todos los días.