I'm a citizen of Nigeria but am living in Italy as an immigrant.
Recently I got the Italian carta d'identita (11 year residence permit).
Can I use this to travel to the US? How do I apply and what else is required?
No. Even Italians can't use a carta d'identita' to visit the US. You need a passport.
The Italian Carta d'Identita for immigrants is different from the one they issue for the Italian citizens; on the cover of the carta d'identita for the immigrants/ foreigners it's written "Non valida per l’Espatrio" which in English means 'not valid for traveling abroad'
The Italian citizens can use the Carta d'Identita within the EU travels, but traveling to the US they do need a passport as well!