Neapolitan (or Nnapulitano) is the Italian "dialect" common to Naples and the surrounding region, one of the most important languages in Italy after standard "Italian" (which was itself originally a Tuscan dialect). The Neapolitan language has long history and rich culture, and those who speak it are justifiably proud of this. It's hard enough for English speakers to learn standard Italian, but if you spend time in Naples it's worth learning a little bit about Neapolitan. After all, many of the most famous "Italian" songs (like O Sole Mio) are actually Neapolitan songs. Also, understanding how Neapolitan sounds will help you understand locals when they talk—even if they are talking in the standard dialect.
Many speakers switch easily back and forth between Neapolitan and "Standard" Italian, sometimes speaking somewhere between the two forms. Many Neapolitans will speak "Standard" Italian with non-Neapolitans, but with a thick accent, which makes it difficult for even native (northern) Italians to understand. Here are some tips on what to expect to help you understand this "Neapolitan accent":
...so, for example, instead of the standard Italian "aspetta" (wait) a Neapolitan might say something that sounds more like "ush-PEHT." Similarly, Ischia is pronounced "EESH-kyuh." Note that this accent is stereotyped in much the same way as some Americans stereotype U.S. Southern accents—as sounding "lazy" or "uneducated."
Neapolitan, like most Italian "dialects" (including "standard" Italian, which is based on Tuscan "dialect"), derives mostly from the simplified versions of Latin that everyday people spoke during the Roman Empire. However, some words betray origins from Greek, since Naples began as a Greek colony (e.g. in Neapolitan parsley is petrusino or petrosino, which comes from the Ancient Greek πετροσέλινον or petrosélinon). Since both the Spanish and the French ruled over Naples over the ages, bits of those languages also seeped in; for example Neapolitan for "yesterday" is ajére, which is similar to Spanish ayer, and Neapolitan for napkin is sarvietta, which is close to French serviette.
Even more fun than Italian proverbs!
The following webpages have more information about Neapolitan.