The group was founded after British forces left Philadelphia and re-established their headquarters in New York City. Washington knew that he needed people on the inside (I've always wanted to use that phrase, and between you and me, I also got quite a kick out of 'rebel-spies') to report on British troop movements. He asked his newly-appointed head of intelligence, Benjamin Tallmadge, to find some spies, which Tallmadge did.
Tallmadge needed people he could trust absolutely, so he began by recruiting his childhood friends. (Nothing says friendship quite like offering someone a job that involves risking their life.) The name "Culper Ring" is taken from "Samuel Culper Sr." and "Samuel Culper Jr.", the code names of two of the ring's main informants. I don't know who they were, but I imagine one was slightly older than the other.
The group's biggest achievement was probably informing Washington of a planned attack on the French forces in Rhode Island in 1780. They're also credited with uncovering Benedict Arnold's betrayal of West Point - though evidence for that is sketchier. In any case, the ring delivered secret messages for years, with none of its members ever being captured (with one possible exception - a woman referred to as "355", whose real identity and fate isn't known.)
The group is credited for being the most effective intelligence operation in the war - on either side. You can't know how much it pains me to write that. It took AGES.