A detective (aka investigator) is an agent who investigates crimes in order to solve them. This can be done on behalf of a government or as one's private occupation. Those who do it privately are known as a private eye, or P.I.
Dixon Hill was a fictional 20th century Earth detective. Captain Jean-Luc Picard was fond of Dixon Hill novels, and in his free time would often escape to the holodeck where he would assume the role of Hill in holographic recreations of the detective's stories. (TNG: "The Big Goodbye", "Manhunt"; Star Trek: First Contact)
Sherlock Holmes was a famous Earth detective whom Data once imitated. (TNG: "Elementary, Dear Data")
In his capacity as Constable aboard Deep Space 9, Odo performed many detective duties for Gul Dukat and later Commander Benjamin Sisko. (DS9: "Necessary Evil")
Miles O'Brien was fond of the Mike Hammer detective novels by Mickey Spillane, including I, the Jury and Kiss Me Deadly. (DS9: "Profit and Loss", "Shadows and Symbols")