The USS Enterprise-C is the fourth starship to be operated by Starfleet by that name and is commanded by Captain Rachel Garrett.
Prime Timeline Edit
The Enterprise responded to a distress call by the Klingon outpost on Narendra III, which was under attack by Romulans. The Enterprise was destroyed in its unsuccessful attempt to help, but the Klingons were impressed by the honor of their act. This helped lead to the eventual long-term peace treaty between the Klingon Empire and the Federation. This almost didn't occur due to a time rift created by the battle that put the Enterprise in 2366, where it created an alternate timeline where the Klingons and Federation were at war. In this timeline, Tasha Yar was still alive due to the different mission profile of the USS Enterprise-D.
The Enterprise-D encountered the damaged Enterprise-C and helped fix her to send her back to finish the battle: a suicide mission to correct the timeline. Since she learned that she would already be dead in the corrected timeline from space witch Guinan, Yar joined the Enterprise-C crew in its return to 2344. (TNG: "Yesterday's Enterprise")
Viggo Timeline Edit
On board the Enterprise-C is a Crewman Daniels who is the uncle of the Daniels that served on the Enterprise-D before transferring to the USS Hood. This Daniels was promoted to acting chief engineer when all of the engineering officers on the Enterprise-C were killed. (SS: "Radio Letters")