Extradition is the legal process of one government requesting the transfer of an individual from another government's jurisdiction to their custody for alleged crimes within their jurisdiction.
When the Federation receives an extradition request, a hearing is held to determine if enough evidence exists to warrant a trial. If it is decided that sufficient evidence exists, the charged individual is turned over to the other government and then is subject to their legal system. The Bajorans handle extradition requests in a similar fashion. (TNG: "A Matter of Perspective"; DS9: "Dax")
William T. Riker was the subject of an extradition request by the Tanugan security force in 2366 when he was accused of the murder of Doctor Nel Apgar in orbit of Tanuga IV. Though initially it appeared as if Captain Jean-Luc Picard would have to grant extradition, due to the evidence, it was eventually found that the Doctor had killed himself while attempting to kill Riker. The security force subsequently dropped its request. (TNG: "A Matter of Perspective")
Jadzia Dax was the subject of an extradition request in 2369 by the government of Klaestron IV, also for murder. This case, was unique in that the alleged murder was committed by the previous host of the Dax symbiont, Curzon Dax. The request was also complicated by the fact that Deep Space 9 was a Bajoran facility, and the Klaestron government had no extradition treaty with Bajor. This required the Bajoran judge Els Renora to hold a hearing. Eventually, an alibi for Curzon was discovered, and the request was dropped. (DS9: "Dax")
The Klingon Empire requested the extradition of Worf in 2372 for the alleged murder of 441 Klingon civilians by destroying their unarmed transport ship during a battle. The hearing, presided over by Admiral T'Lara, eventually found that the ship Worf ordered destroyed was empty, and the allegedly dead civilians had died in a separate accident. The Klingons hoped to stop Federation aid shipments to the Cardassians with the embarrassment of the incident. (DS9: "Rules of Engagement")