Italy has a dual approach of handling emergency calls: it has both specific telephone numbers for medical (118), fire (115), police (113 and 112), car problems (116) and other emergency services, and a "unified" emergency number, which appears to be shifting from 113 to 112 in the longer term. If this sounds a bit confusing, keep in mind that Italians are masters of improvisation (which can be helpful when the unexpected happens), but they are not famous for long term planning...
While in Italy you can dial 118 for medical emergencies (Udine EMS ranks among the finest in Europe). This is the one dedicated number which you should remember, as it can save precious time.
Italy has two police forces, the State Police (Polizia, under the authority of the Ministry of Interior) and the Carabinieri corps (Ministry of Defense), which respectively run the 113 and 112 emergency telephone services. 113 used to be the number to call for various types of emergencies, with the operator handling and/or forwarding the call as appropriate for non-law-enforcement services. Then came new special numbers (e.g. 118), as well as international standards where 112 was preferred as a unified emergency number (it is probably the best three-digit number for both touch tone and rotary phones). The success of GSM cellphones implementing 112 as a special number (which can be called on any network without unlocking the phone and even without a valid Subscriber Identity Module) helped make 112 accepted as a general purpose emergency number. In practice, 113 is still the best general purpose number, even if 113 and 112 increasingly often end up being answered by the same people.