The area of Rome known as the “centro storico,” or “historic center,” is sometimes referred to as whatever lies inside the ancient Aurelian Walls, but the border the walls created aren't exactly the same as what many people refer to as the Centro Storico today.
UNESCO designated the “Historic Center of Rome” a World Heritage Site in 1980, declaring the area inside the Aurelian Walls plus Vatican City (which was outside the walls) to be the city's Centro Storico. To most visitors, however, the Centro Storico is much smaller, and where many of the main attractions are located.
In the Centro Storico, you can visit sights such as the Pantheon, Piazza Navona, Trevi Fountain, and the Capitoline Hill. The Forum and Colosseum are just outside the smallest interpretation of the Centro Storico, as are Vatican City and the Trastevere neighborhood.
Some say the Centro Storico roughly means the same thing as the area formerly known as the Campus Martius – that's the smallest version of the Centro Storico. When you're talking to a taxi driver, however, they use the old border of the Aurelian Walls to determine fares.