wait what? I always thought "capeesh" was just "capisce" with the end swallowed? Is "capisce" not standard italian?

Capisce, pronounced with a distinct 'eh' at the end ('capeesh-eh'), would be standard Italian for 'he/she understands' or 'you (polite) understand'.

But 'capeesh' tends to be used differently in American mob films, meaning either 'Got it, pal?' or 'Yeah, I got it' ("Capeesh? Capeesh."). Those would be different in standard Italian: capisci ('capeesh-ee'), and capisco ('cap-is-coh'), respectively. It's that final example that makes it obvious that the mobsters aren't speaking standard Italian - there is no 'sh' sound in capisco, so eliding the final vowel wouldn't get you to 'capeesh', but to something more like 'cap-isk'.

However, the corresponding forms in Sicilian are capisci and capisciu. Eliding the final vowel yields the observed 'capeesh' in both cases.

It makes perfect sense that the mobsters would be speaking Sicilian rather than standard Italian. Italian immigrants in the US were overwhelmingly from Italy's south, which is generally poorer than the north. (The Mafia, in particular, is an organization with its roots in western Sicily.) Most of these immigrants came before the advent of standardized/centralized schooling in Italy, and so were never taught modern standard Italian. Instead, they spoke their native Romance languages, generally dialects of Sicilian and Neapolitan.

Even today, most Italian-Americans will be able to tell you which 'dialect' their grandparents spoke.

Standard Italian: capisci

   [Capisce is] borrowed from the spoken Sicilian and Neapolitan equivalents of Italian capisci, the second-person singular present indicative form of capire (“to understand”).

No, it is not. When they say "capeesh" in the movie, they're trying to say "do you understand?" (second person singular). In Italian, that would be "capisci" (pronounced "ca-pee-shee").

Additionally, capisce ("does he understand?") in Italian, is pronounced "ca-pee-sheh".

The "capeesh" is derived from Sicilian.