The term cycle was an unspecified unit of time used by difference species.
When Thylek Shran posed as a member of the Andorian Mining Consortium in 2153, searching for archerite, he told Degra that what he sought was extremely valuable to the Andorians, and that "[e]ven a few kilos will pay our expenses for the next ten cycles." (ENT: "Proving Ground")
The Borg measured their maturation and regeneration cycles by individual units. According to Seven of Nine, assimilated children were placed in the chamber for a period of seventeen cycles. (VOY: "Mortal Coil") Later, when Seven of Nine discussed Icheb's regenerative needs to Leucon, she explained that she adapted a neural transceiver to interface with a portable regenerator, and figured the device had enough power to complete one full cycle for Icheb. Though the length of a cycle was not specified, she did add that "[i]t's important that Icheb regenerate for six uninterrupted hours" per day. (VOY: "Child's Play")
The Romulan officer Decius, aboard the Praetor's flagship, noted in 2266 that they had no motion on their sensors for twenty cycles. James T. Kirk had also noted in a log that the USS Enterprise had been motionless for nine hours and 47 minutes. (TOS: "Balance of Terror")
The Argrathi cycle was equivalent to one Earth year. The crime of espionage required a minimum of fifteen cycles of correction on Argratha, and Miles O'Brien served twenty cycles, the equivalent of twenty Earth years (though his entire imprisonment was actually an artificial memory). (DS9: "Hard Time")