Modern Rome is divided into four counties and 155 districts. You're likely to be interested in the Old Town, or Municipio I, which is divided into 22 districts. However, it's more of a formality - you're better off walking around the city in accordance with the tourist districts on the map of Rome.
For example, the centre (Spanish Steps, Pantheon, Colosseum, Piazza Navona), the Vatican City (if Christmas or Easter is far away, you can rent an inexpensive apartment there) or Trastevere (the 'soul' of Rome, with its many trattorias serving the tastiest pasta).