First things first, S.H.I.E.L.D. stands for Strategic Hazard Intervention, Espionage and Logistics Directorate.
S.H.I.E.L.D. was formed by an Army colonel named Nick Fury after World War II.
Artist Jim Sterenko started doing art for Nick Fury and S.H.I.E.L.D. in 1966, and the series became known for Sterenko's striking, psychedelic style.
The Nick Fury of the comic books is not the Nick Fury you know and love from the films.
The original Nick Fury was, most notably, a white dude. Like Samuel L. Jackson's Nick Fury, the comic book Fury is missing his left eye. In fact, pretty much every Nick Fury is missing that left eye.
David Hasselhof once played him in a made-for-TV movie called Nick Fury: Agent of S.H.I.E.L.D..
There are other Nicks Fury as well.
The most notable is the Nick Fury of the alternate "Ultimates" universe, who was designed to resemble Samuel L. Jackson. Marvel also recently introduced Nick Fury, Jr. (son of the original Nick Fury), who was also created to have a comic book analogue for the popular Nick Fury character of the film and TV universe.
There are also a ton of other versions of Nick Fury, because comic books are crazy and full of alternate dimensions and timelines.
(Not a single left eye to be found.)
In the comic books, Nick Fury started S.H.I.E.L.D. and ran it most of the time, but some other notable characters have held the reins as well.
Fury was in charge of the organization up until he started a secret war this one time. After Fury left command, a number of others controlled the organization for a while, including Tony Stark (aka Iron Man), Norman Osborn (aka Green Goblin), and Maria Hill (who was played by Cobie Smulders in the films.)
Writer Jonathan Hickman implied that S.H.I.E.L.D. even has roots all the way back in the 16th century, and that Leonardo Da Vinci and Michelangelo were among its founding members. (Also, Michelangelo is a glowing, time-jumping super-being, so...)
In the film universe, it's not clear how S.H.I.E.L.D. gets started, but Tony Stark's father, Howard Stark, is involved in its creation.
Agent Phillip Coulson (played by Clark Gregg) started his fictional life as a creation for the film universe, but was recently added to the comic book universe as well.
In the comic books, Agent Coulson fought alongside Nick Fury, Jr. in the Army. Like his film counterpart, the comic book Coulson was responsible for assembling a team of the Avengers. (Unlike the film Coulson, however, the comic book Coulson was only responsible for creating a team known as the Secret Avengers, not the Avengers Avengers.)
Marvel's Agents of S.H.I.E.L.D. will feature Agent Coulson (who has miraculously returned from his apparent death in The Avengers) and a number of agents who are new to the Marvel universe.
Marvel's Agents of S.H.I.E.L.D. will pick up where the Marvel film universe left off in The Avengers and Iron Man 3.
The S.H.I.E.L.D. of the comic books has had some notable members of their own, including Dum Dum Dugan and Gabe Jones, both of whom made appearances in Captain America: The First Avenger.
Both characters got their start in Nick Fury's Howling Commandos during World War II. Like most comic book characters, both have made it to the modern era due to crazy circumstances like time travel and cryogenics.
S.H.I.E.L.D. is also responsible for a lot of pretty interesting technology in the Marvel Universe, such as their signature Helicarrier, which you saw in The Avengers.
S.H.I.E.L.D. also pioneered the technology behind Life Model Decoys, and is capable of creating robotic duplicates of people.
S.H.I.E.L.D.'s success has led to a lot of off-shoots and imitators. Here are a few:
Each group serves its own purpose. A.R.M.O.R. was formed to fight paranormal threats in the Marvel Zombies series, H.A.M.M.E.R. was formed while the villain Norman Osborn was in charge of S.H.I.E.L.D., and S.W.O.R.D. is responsible for monitoring threats from space and extraterrestrial planets.