The Normandy is a travelling Freak Show that flies around the Galaxy entertaining the masses. Its crew of freaks saved the Galaxy from MC Sovereign's attempt to bring his buddie back and was turned into a warship shortly after. Despite the fact that they are in the same business, the Normandy is not a direct competitor of either Devan's Travelling Circus or the Circus of Wonders due to the fact that it is simply a freak show, it provides not entertainment of its own, people simply pay to see the many attractions on offer.
The Normandy SR-1 was the ship that delivered the final blow to MC Sovereign as he tried to jack the Citadel's antennae in order to broadcast his tunezzzz throughout League Space, while The Crew on the ground confronted the madman Saren in an epic battle at the top of the Citadel Tower. However, it was destroyed by an MC Sovereign fan cult piloting a Giant Piece of Turd in 2183. Despite the fact that the ship was blown to bits, all of the crew but two survived. And one of the guys that died was reconstructed from a broken pulp of flesh into a fully functioning person again. Somehow.
The Normandy SR-2 was built by a Pro-Human terrorist group (Wow! Pro-Human?) called Cerberus. The new version engaged in battle against Harbinger and his legion of creepy insects in 2185. During the attack on the insect's base, the SR-2 destroyed the Flying Piece of Turd in two shots and then demolished the base of fanboys, ending the legacy and chart domination of MC Sovereign.
Following the Reaper War, the Normandy SR-2 was reccommisioned as a Freak Show again and re-recruited its old crew. (Including the dead ones, who were reconstructed using the same method stated above). The Normandy now travels all over League Space, offering spectacles never seen before to citizens of the Galaxies.