The favorite area for visitors to Rome is the historic center and the area near Termini station. Most of the attractions are located in this area but being well connected by metro and bus it is not compulsory to stay in this area.
The area of the Trevi Fountain, Piazza Navona, Pantheon and Campo de 'Fiori is the most sought after to stay in Rome. This whole area is easily explored on foot and is not far from the Colosseum and Roman Forum area.
Beware that hotels in this area may cost more than the quality offered.