A cycle is a unit of time measurement used by both Romulans and Argrathi.
Romulan cycle
One Romulan cycle is approximately equivalent to no more than 29.35 Earth minutes. A Romulan officer noted they had no motion on their sensors for 20 cycles. Kirk had also noted in a log that the Enterprise had been motionless for 9 hours 47 minutes. (TOS: "Balance of Terror")
Argrathi cycle
An Argrathi cycle is 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 implanted memory). (DS9: "Hard Time")
- In the background chatter of TNG: "Unification II", it is possible to hear dialogue noting something about "two cycles". The exact line is "We are very close now... within two more cycles."