Crimson Team participated in various operations against Covenant and Promethean deployments on Requiem. They were then sent to shut down a Covenant archaeological dig.
They were then deployed in tandem with Fireteam Majestic to clear out Forerunner structures. Crimson cleared their structure first, much to the annoyance of Majestic. As the battle of Requiem raged on, fireteam Crimson managed to assassinate Parg Vol, a Sangheili terrorist. Later, in pursuit of Jul 'Mdama they managed to secure a Forerunner device known to the Covenant as The Didact's Gift.
They were later captured by the Covenant on a search and rescue op for Gabriel Thorne but escaped, and began a series of crippling surprise attacks on the Covenant using a Phantom flown by LieutenantTJ Murphy before being recalled to Infinity, which was under attack. After crashing in the hangar, they disarmed several stolen HAVOK Tactical Nuclear Weapons, turned the rear outer defenses back online, and thus secured the Infinity.
They later were tasked by Infinity to assist with the deactivation of one of the two Artifacts that was holding Infinity in Requiem's gravity field. After succeeding, they raced to Infinity in a Pelican while Requiem was pulled into its own sun. After 'landing' on the massive ship, Requiem was destroyed, while Infinity jumped into slipspace.
It is quite possible that 343 named the fireteam after Bungie's game, Crimson: Steam Pirates.
If you pay close attention to Spartan Ops Episode 5 Chapter 5 end cutscene, you can notice your own Spartan Color and Armor (specially if playing alone), suggesting that Spartan Ops in-game cinematics are working with a similar engine to the engine used on Halo Reach.