Sometimes hotels in Rome have private garages, but of course these are the more expensive luxury hotels. Usually they do not have more than a couple of spots available anyway.
Most smaller hotels and bed and breakfasts have an agreement with a garage whereby their guests get a discount. Therefore, always ask your hotel receptionist if they have some such agreement before leaving your car somewhere.
The rates of Roman garages vary, depending on location and time of year (and on how greedy the owners are). A normal rate would be around 30 or 35 Euros per day, but in the center you can also end up paying twice as much.
It is also recommended to ask what the price is without specifying the number of days you wish to park somewhere. After that you can ask for a discount based on the number of nights you will be staying in Rome.
Especially if you have a car with a foreign license plate it is recommended to find a safe, guarded place to park.
Contrary to what you might expect, it is often cheaper in summer, when the Romans themselves are on holiday and there is no commuter traffic in the city.