The Nonnberg is a cliff of the Festungsberg / Mönchsberg hills and often considered to be a separate hill. This is rather peculiar and to be explained by historic rather than geological (or other logical) reasons. Effectively, Mönchsberg, Festungsberg and Nonnberg are one hill.
The Nonnberg ("nun mountain") was named after the nunnery of Stift Nonnberg, which in turn was named after the mountain. It is all very confusing and a bit of a chicken-egg thing. In any case, the nunnery was founded by St. Erentrudis in the 7th century and endowed with the Nonnberg hill and the land in today′s district Nonntal (meaning "nun valley"). The Nonnberg was the site of a small castle, the "Salzpurch" (salt castle) that can be tracked between the 7th century, when Duke Theodo II of Bavaria gave it to St. Ruprecht, and 1057, shortly before the proper fortress was started to be built on the neighbouring Festungsberg.
If you approach the nunnery from the Old Town via the path to the fortress, you will pass the "Nonnberger Hund", a battered Romanesque statue of a lion. This statue served to mark the onset of the nun′s territory. An alternative route to the Nonnberg leads via a staircase from the Kaigasse; on the southern flank of the hill, you can approach the nunnery via the Nonnberggasse. All three routes are very nice walks with great views on Salzburg and the surrounding Alpine scenery.
Website of Stift Nonnberg, with picture