There are different ways to qualify for Italian citizenship, even if you were not born in Italy or have never set a foot there. Due to special agreements with the U.S. and other countries, it is possible to have dual citizenship (USA and Italy) without this affecting either.
The simplest way to gain Italian citizenship is through your paternal ancestors. If any of them was born in Italy and if none of his descendants ever renounced the Italian citizenship, you are entitled to apply for it yourself. There is no limit on the number of generations when you're applying on the paternal side of your family.
If your mother is the one with Italian heritage, rules are slightly different. For you to be able to claim Italian citizenship, you need to have been born after January 1, 1948. All other rules, including not having renounced your right to the citizenship, do apply.
If your parents (rather than your grandparents) are Italian citizens, you are entitled to citizenship yourself through the jus sanguinis principle. This is an automatic adjudication of citizenship and requires little paperwork. On the other hand, citizenship is not automatic for children born in Italy, unless both their parents are Italian. Children of foreign-born people who don't have the Italian citizenship themselves can only be granted citizenship after three years of residency in the country.
Those who marry an Italian citizen can obtain Italian citizenship after two years of legal residency in Italy (up until July 2008, the residency requirement was just six months) or three years of living together in a foreign country. Before 1983, the adjudication of citizenship was automatic following marriage, but due to the number of fake marriages, the requirements are now much more strict.
For people with no family connection to Italy, the only other way to obtain citizenship is through naturalisation. This can only be done after 10 years of residency in the country, as well as proof of financial means and a clean criminal record.
Anybody applying for Italian citizenship will need to present birth certificates of parents and grandparents, as well as their own. Marriage and death certificates are also required. All documents will need to be translated into Italian and require an Apostille. If any of the people in the line of descendants became a naturalised citizen of the U.S. or other country, records showing that they never renounced Italian citizenship will need to be presented as well.