The city of Rome, at least the parts that include the things tourists want to visit, is really rather spread out. Instead of there being a compact and historic city center that is easily explored on foot, as in many other old European cities, many of Rome’s sights are far enough apart that trying to find a cheap hotel in Rome that is within walking distance of everything you want to see is nigh to impossible.
Not only that, Rome is a year-round tourist destination, making it tough to find cheap hotels in Rome at any time of year. Many of Rome’s budget hotels are clustered around the Termini train station, which is not a central location but is well-connected by mass transit (both Metro and bus lines). The important thing to keep in mind when looking at hotels near the train station are what side of the station you are on – some people feel that the southwest side is less than desirable, especially at night. The northeast side tends to feel much less questionable, even after hours. If you want to stay somewhat closer to the sights, be prepared to pay more.
Some budget hotels to consider for your trip to Rome are: Hotel Cervia, Hotel Positano, Hotel Continentale (all located on the northeast side of Termini station on Via Palestro), and the Hotel Santa Prassede (a short walk from the Spanish Steps).