The Reaper War was a war fought in the years 2185 PLF -
2187 PLF. It was the attack by The Reapers on the Galaxy in an attempt to wipe out all advanced sentient life. After leaving the League nearly 2400 years earlier, after the conclusion of the First Galactic War, the Reapers had been waiting in the Dark Space for a chance of revenge.
Many planets were attacked or destroyed during the war. The Turian homeworld of Palaven was razed, the Quarian Migrant fleet was heavily damaged and the Asari and Salarians lost several fleets too. The Turian Dreadnought Indomitable spearheaded an attack on the Reapers in defence of Palaven, the Turian homeworld, which turned the tide of the war and allowed other races (such as the Asari with their flagship, the Destiny Ascension) to sweep in and rout the Reapers on many planets in the Galaxy. However, this did not save the many planets conquered by the Reapers from suffering massive damage. Funnily enough, the most damaged planet just happened to be the Human homeworld of Earth. Ironic, considering it seems to be attacked and almost destroyed during every single war Humans are involved in.
It took the combined forces of all of the races in League Space to eventually defeat the Reapers. They also had help from The Composer, which was called The Crucible at the time.
After The Reapers were defeated, they suddenly decided to turn nice and help everyone rebuild. They rebuilt Earth first to make up for the amount of damage they caused to it, then went on to rebuild the rest of the Galaxy.
It was this dedication to rebuilding, and the protection and security The Reapers offered to the citizens of the Galaxy, that made them an important part of the League and its military after they re-joined in 2192 PLF. Harbinger also played an important part in convincing Gravemind to join the league nearly 400 years later.