The Albaicin (also spelled Albayzin or Albaycin) is Granada's old Muslim quarter, and its steep twisting streets still have a medieval feel. With its white buildings and deep-gardened mansions spilling down the hill, the Albaicin is beautiful in itself, but what makes it particularly stunning is its views of the Alhambra. (The views of the Albaicin from the Alhambra enhance that experience as well!) There's a viewing point by the church of St. Nicolas that offers particularly good Alhambra vistas.
The Albaicin was heritage-listed in 1984. Its name may have derived from settlers fleeing the Christian invasion of the town Baeza, or it may derive from an Arabic phrase meaning 'quarter of the falconers.' Despite the Christian conquest of the city in 1492, it survived as a Muslim quarter for some decades, and you can still see the remains of Islamic bathhouses, mansions and fountains.
Buses 31 and 32 from Plaza Nueva will take you on a circular route around the Albaicin. They run roughly every 10 minutes.