Inside the old city walls of Florence (well, what used to be the walls and is now the viale or large avenue circling the city), driving is limited to registered vehicles (city residents) except for a small area around the train station (Santa Maria Novella). Florentines call this the ZTL, or "Zona Traffico Limitato". It is important to know this to avoid receiving HUGE traffic tickets by driving your car into the center of Florence. On the map below (click it for a larger version), you can see the ZTL entry points (red and blue "T"s) and the various zones by letter (which correspond to where residents can park). You can drive into the center with a rental car if you are staying at a hotel, but the hotel must let the traffic police know your license plates, etc. And if you do drive in, make sure to use one of the blue "T" spots, not the red - even with a residents permit to enter, you can't enter at the red "T" signs (I now know - it was an expensive lesson).
So remember - if you are just coming to Florence as stop on a day trip, or you are staying at a self catering apartment (i.e. not a hotel), you can not legally drive into (most of) the center, although personally we have heard stories of people doing so and not getting tickets, if you do the fines are steep and they are getting better and better at finding you and making sure you pay. In any case, this is something to be aware of and be careful of! Study the map closely and you will see there are a few ins and outs (Sundays are open, as are nights from 7:30 PM to 7:30 AM, but not in the summer, etc.). Lastly, the light blue shaded streets are "zona pedonale" or pedestrian zones, and you shouldn't be driving on them at all!