Panoramic view from Trinità dei Monti: view over piazza di Spagna, via dei Condotti, with the frenzy of the weekend for shopping!
Panoramic view from the Pincio terrace at night: you can see piazza del Popolo and St Peter's cupola.
View from the cupola of St Peter's basilica. You have to pay to get in, but it's really fantastic (not to mention all the rest you see after paying for the entrance at the Vatican's museums)
View from the Gianicolo hill. Despite not as central as some of the other places listed here, the Gianicolo is one of the most popular panoramic spots in Rome.
Via dei Fori seen from the back of Campidoglio hill. In the weekend you'll see a river of people walking and enjoying Rome's atmosphere.
The existence of the panoramic terrace on top of the Altare della Patria, reachable with the glass lift, is not known by many tourists, despite the location in the heart of Rome, and the spectacular view in many direction.
The belvedere dello Zodiaco, which takes its name from the homonymous restaurant, is on the highest hill in Rome: monte Mario. This place is one of the most popular couples destinations in my city.
The Orange garden has a lot of interesting things to offer as it's located in an area rich of tourist gems, among these there's surely the beautiful view from the terrace.