The Heuberg is a hill stretching from the Gaisberg towards the city of Salzburg. By some measures, the Heuberg is one of the Stadtberge (City Hills) of Salzburg. It is 901 metres high and shared between the municipalities of Salzburg, Eugendorf, Hallwang, Plainfeld and Koppl. It is home to a very exclusive settlement of villas and offers nice views on the city of Salzburg and its surroundings as well as the Alps. There are well-known inns that benefit from the prominent location of the Heuberg.
Settlements on the Heuberg can be tracked back to the early Middle Ages (thus not all that far for an area that has often been populated since the Stone Age). The gentle landscape was inappropriate for the construction of sheltered communities, but the suitable for agriculture once the forest had been cut down. The name "Heuberg" (Hay Mountain) refers to that and was first recorded in 1141. The forests in the northern parts of the Heuberg belonged to the monasters of Stift St. Peter for centuries (partly, they still do).
Today, the Heuberg is generally associated with wealthy people who own villas there. However, the area is also in important recreational region for locals and popular for walking, hiking and gentle winter sports such as cross country skiing. The Heuberg is within easy reach from the city if you have a car, public transport is a pain, though. If you are keen on easy hiking and limited to public transport, then the Gaisberg might be a more rewarding destination for you.
German Wikipedia on the Heuberg