(covers information from several alternate timelines)
The crime of conspiracy is the act of collaborating with other persons to advance another criminal activity.
From 2258 of the alternate reality, Section 31, led by Admiral Alexander Marcus, conspired to militarize Starfleet for war against the Klingon Empire. Marcus recruited the war criminal Khan Noonien Singh under the name of John Harrison to design ships and weapons. However, Khan fled by the following year when his plan to smuggle out his fellow Augments was exposed, leading to his vengeful attacks on the Kelvin Memorial Archive and Starfleet Headquarters.
Marcus attempted to cover up the conspiracy by ordering James T. Kirk to execute Harrison with the photon torpedoes he had attempted to smuggle his crew in. However, Kirk learned the truth when he chose to arrest Khan instead of bombing him from orbit. Marcus came in the USS Vengeance and when it became clear Kirk wanted to bring Khan to trial and expose the conspiracy, he attempted to destroy the USS Enterprise, but was prevented from doing so. (Star Trek Into Darkness)
In 2286, the crew of the USS Enterprise, led by Kirk, was charged with conspiracy after their retrieval of Captain Spock from the Genesis Planet. The conspiracy, as charged, led to acts of assault, sabotage, theft, and destruction of Federation property. All of the charges except for one were summarily dismissed by the President of the Federation. (Star Trek IV: The Voyage Home)
In 2364, a race of neural parasites infiltrated Starfleet Command. Admiral Gregory Quinn was the first to suspect a conspiracy was going on at the highest levels of Starfleet. The alien plot was foiled by the efforts of the crew of the USS Enterprise-D and Admiral Norah Satie, among others. (TNG: "Coming of Age", "Conspiracy", "The Drumhead")
In 2369 the senior staff of Deep Space 9 talked about a way to arrest the Bajoran Kohn-Ma terrorist Tahna Los. Doctor Julian Bashir proposed that they could charge him of conspiracy and Constable Odo told that conspiracy is a crime. (DS9: "Past Prologue")
The same year, Commander Sisko was able to prevent an assassination attempt on Vedek Bareil Antos. When he talked with Major Kira Nerys after this incident and the interrogation of the assassin Neela, both believed they couldn't prove any connection to Vedek Winn Adami and Neela was the only one who was accused of conspiracy. (DS9: "In the Hands of the Prophets")
In 2377 Kadan conspired with the Quarren Ministry of Health and officials from Quarren Criminal Investigations to abduct aliens to relieve a massive labor shortage. This also included altering their memories so the victims would not realize their situation, and actually be happy to be working on Quarra. USS Voyager personnel were victims of this conspiracy, as well as numerous others. After being discovered, the Quarren ambassador promised that all the victims would be freed. (VOY: "Workforce", "Workforce, Part II")
In an alternate 2390, Captain Geordi La Forge told the renegade Chakotay that charges of conspiracy to violate the Temporal Prime Directive would be dropped if he, Harry Kim, and Tessa Omond surrendered themselves and a Borg temporal transmitter in their possession. Tessa pointed out that such an offer had little meaning to them, as if their plan to prevent the USS Voyager from crashing in the Takara sector was successful, the timeline would be changed and the charges would not exist. (VOY: "Timeless")