The compound shown is protonated by acid. At which nitrogen does protonation occur, and why?

Protonation at the ring nitrogen (pKaH about 9.6, far above pyridine at 5.2) gives a cation in which the NMe2 lone pair is conjugated with the ring N+-H (amidinium-like resonance, positive charge on NMe2 nitrogen). Protonating NMe2 would remove this donor and give a poorly stabilised ammonium, so NMe2 is the less basic site.