Foreign students must have a healthcare coverage to be able to stay in Italy.
If you are an EU citizen, The New European Health Insurance Card gives access to immediate necessary care under the same conditions as nationals of that country. Anyway, if you are planning to stay in Italy for more than 3 months, you may be required a paper form (such as the S1 form).
If you are not an EU citizen, you must have a health insurance in order to obtain the residence permit to stay in Italy.
You can take out a private health insurance, subscribed in your Home Country or in Italy, but please note that not all private insurances are accepted for residence permit applications, so you should check that the insurance you choose is recognized.
Alternatively, You can register with the National Health Service (SSN) (GUIDEBOOKS ARE PROVIDED IN SEVERAL LANGUAGES). Please note that some Countries have special reciprocal Health Care Agreements with Italy (you can check the list here, only in italian)