The iconic film private eye is probably Humphrey Bogart, who played Sam Spade in the classic 1941 version of The Maltese Falcon and Philip Marlowe in 1946's The Big Sleep. Both Spade and Marlowe are classic characters from pulp detective stories. These characters, and more specifically Bogart's take on them, did a lot to set in place the image of the P.I. that you're talking about, so they're a great place to start.

The P.I. movie is part of a broader genre called film noir, whose name refers both to its visual style and use of light and shadow (emphasized by the black-and-white film most noirs were shot on) and to the cynical, hard-bitten plot of the typical noir story. Once you've got up to speed on P.I. movies you may want to explore the rest of the noir genre as well; good places to start there are 1944's Double Indemnity, 1949's The Third Man (a British noir, which contrasts interestingly with the American version), and White Heat, also from '49.

There have been a lot of attempts at "neo-noir" since the genre's classic period ended in the 1950s, but they tend to be hit or miss because many neo-noir filmmakers focus more on the visual style than on the underlying sense of futility and hopelessness that noir embraced. The best of them is probably Chinatown, from 1974. Blade Runner from 1982 is a classic sci-fi adaptation of the noir style. For a decent (and underappreciated, sadly) modern take on the figure of the noir-style P.I., look up 1998's Zero Effect.