The Exodites are the Eldar who first fled the decay and degeneration of their old Empire, effectively being the Eldar equivalent of the Amish. They maintain close bonds with the Craftworlds on whom they are heavily dependant for protection. They're basically Amish dinosaur-cowboy Wood Elves In Space, and very like the Na'vi from James Cameron's Avatar. They also have Eldar Knights.
The Exodites originated from those citizens of the Eldar Empire who saw the corruption festering in the souls of the Eldar and decided they needed to purify themselves. To do so, they left the Empire and its conveniences for worlds on the edge of the galaxy (an exodus, hence "Exodites"), where they would be forced to work hard just to survive. This kept them focused and busy, and prevented them from being lost in hedonism like their relatives back home. When the Empire's increasing debauchery gave rise to Slaanesh, the Exodites survived because they were far from the center of the Empire (what would later be called the Eye of Terror) and because their hard lifestyle and self-denial meant that Slaanesh had little purchase on their souls. Of course, the Eldar idea of "hard lifestyle and self-denial" means that they have to walk a few feet to pick up their own damn food instead of it appearing before them automatically, though it still grows flawlessly for them and every plant and animal on the planet caters to their every need and whim. Apparently, they are not completely free of Slaanesh's influence; when they die, their souls are stored and contained in their planet's World Spirit (analogous to the Infinity Circuit of the Craftworlds).
As of the 41st Millennium, the Exodites still keep to themselves, though the expansion of the Imperium of Man and the encroaching Tyranid Hive Fleets have brought them into conflict. They defend themselves with their dinosaur-riders, and Craftworld Biel-tan often dispatches forces to aid Exodites under attack. Though, in Black Library books Exodites are described as absolute badasses and comparable to the Catachan, so they likely do not need any help with beating the shit out of anyone stupid enough to invade their world. In one book they almost effortlessly repelled four Kabal raids combined, and in another,
one entire THREE pre-heresy Space Marine Legions were needed to conquer a single fucking Exodite world. That's kind of badass, even though the Imperium at the time arguably had more advanced equipment than what the Exodites denied to themselves. It should be noted that the only reason for this is not due to any particular skill of the Exodites, but rather, as depicted in the book "Deathworld", an Exodite world becomes Catachan on crack towards any invaders. The Exodites need only hide and perform hit-and-run attacks on invading convoys and supply depots while letting the planet itself do the rest. Of course, there's a fine and dandy way of solving that little issue, but that isn't always an option.
In many pieces of more recent literature, however, whenever the Exodites show up, they are GW's second favorite punching bag right after the Craftworld Eldar. Many of the battle reports found in the codices and some novels show that Exodite worlds get sacked and burned by a single strike force of Orks or Space Marines. Yeah.
In general Exodites are only mentioned when they die, and rarely do Exodite worlds not appear unless some Ork force or Space Marine Company is wiping them all out.