The Cypherpunks movement coalesced around the late ‘80s and early ‘90s shortly after public-key cryptography came into the public eye in the late ‘70s with Diffie-Hellman and RSA, and then PGP in ‘91. The movement mainly consisted of libertarians and privacy-enthusiasts who wanted to use cryptography’s “defender’s advantage” to maintain online privacy and create anonymous communication systems to circumvent surveillance and censorship.
Grasping cryptography fundamentals as well as an anti-government intuition is key to appreciating the Cypherpunks and their influence. The movement took off in 1992 with the popular Cypherpunks Mailing List and SF monthly meetup, created and led by Tim May, Jim Gilmore, and Eric Hughes.
The following are recommended reads on the Cypherpunks.
Security Without Identification: Transaction Systems to Make Big Brother Obsolete [David Chaum 1985] Cryptography basics; pseudonymous transactions while maintaining security for all parties. The technical roots of the movement are traced back to Chaum and the ideas in this paper.
From Crossbows to Cryptography: Thwarting the State via Technology [Chuck Hammill 1987] A call for libertarians to embrace cryptography as a robust defense against the state, and to spread information about this technology, as information markets are not zero-sum and can multiply one’s efficacy. A transcript of this speech was the first message on the mailing list.
Cypherpunks Mailing List archives [1992-1998] Somewhat dense and hard to parse through in the archived format, but Wikipedia says this was an “active forum with technical discussion ranging over mathematics, cryptography, computer science, political and philosophical discussion, personal arguments and attacks, etc., with some spam thrown in.”
A Cypherpunk’s Manifesto [Eric Hughes 1993]
Crypto Rebels [Steven Levy 1993] Wired article; first mass media discussion of cypherpunks. [There is also a very good book by Levy on the subject – “Crypto”]
Cyphernomicon [Timothy May 1994]
John Gilmore’s home page [last updated 2013]
The Crypto Anarchist Manifesto [Timothy May 1992]
Assassination Politics [Jim Bell 1997] Probably the most read Internet essay in history. Bell is an important figure in the crypto-anarchy and anarcho-capitalist movements and was the first to detail this idea of anonymous assassination markets.
A Declaration of the Independence of Cyberspace [John Perry Barlow 1996]
A CryptoParty Manifesto 
DEF CON archives [1993-2016]