A time-active civilisation or power was one which had mastered time travel and its paradoxes. These were apparently not common in the universe, in no small part due to the intervention of the Time Lords, who did not look kindly upon potential rivals. (TV: The Time Warrior, The Two Doctors, PROSE: Sky Pirates!)
Examples of time-active
The Faction Paradox was a time active organisation which revelled in paradox. They could also be seen in some fashions as a temporal terrorist organisation. (PROSE: Alien Bodies)
The Daleks had time machines, (TV: The Chase and others) and used time corridors (TV: Resurrection of the Daleks) and emergency temporal shifts. (TV: Doomsday)
The Monan Host, while not on par with the Time Lords, possessed powerful time ships. (AUDIO: The Apocalypse Element)
The Enemy were another power opposed to the Time Lords. (PROSE: Alien Bodies)
The Time Agents in the Time Agency used vortex manipulators to master time/space travel. (TV: Utopia, The Sound of Drums)
The Trods developed time travel technology. (COMIC: Pursued by the Trods)
The Cybermen used time travel on several occasions in an attempt to conquer more vulnerable time periods. (TV: Attack of the Cybermen, AUDIO: Real Time, COMIC: The Flood)
The Androzani trees were capable of forming a psychically controlled room capable of safely navigating the Time Vortex. (TV: The Doctor, the Widow and the Wardrobe)
A group of Slitheen managed to create temporal drives for use in a time travel tourist package. (PROSE: The Slitheen Excursion)
The Korven had ambitions to rule time and space and attempted this by invading in the past planets they had failed to conquer in their own time. (TV: The Korven, The Eclipse of the Korven)
The Nekkistani used time-cruisers to travel the Time Vortex. (AUDIO: Terror Firma)
The Order of the Black Sun started the first time war against the Time Lords. (COMIC: Star Death, 4-D War, Black Sun Rising)
Other references
After enduring the Cybermen incursion into the Federation universe, as well as encountering the Eleventh Doctor, the Borg sought to investigate and master time travel in order to assimilate the Time Lord. (COMIC: Assimilation²)
Behind the scenes
The film Star Trek: First Contact, which was released beforehand, establishes that the Borg did indeed master time travel as they intended to at the end of Assimilation², with the ability being demonstrated by a Borg sphere. While the Vulcans initially deemed time travel to be impossible, the Star Trek canon establishes that both the United Federation of Planets and the Klingon Empire developed time travel capabilities prior to Assimilation², though the crossover does not acknowledge this.