An ecclesiastical Latin translation of the Greek word exarkhos, ("ruler out of"), the term "exarch" is both a rank used in the Orthodox Catholic Church and a historic term for a governer in the Byzantine empire. When it comes to /tg/, however, there are two distinct uses for it.
In Dungeons & Dragons 4e, the term "Exarch" was brought in as a reference for powerful servitors of the gods. This catch-all terminology simplified the oft-Byzantine array of demigods, lesser gods, godlings, godspawn, tribal heroes, saints, proxies, archangels, and other such individuals who directly served the gods whilst being almost, but not quite, equal to them.
This was used to downgrade many lesser deities who didn't really have a huge impact on the typical adventuring party. Whilst mainstream deities would have broad portfolios, more aspect-focused deities would serve under them. For example, many of the dwarven and elvish pantheons became exarches subservient to their traditional "Greater Gods", Moradin and Corellon respectively.
Beyond the whole "epic-level servant of the gods" thing, exarches were a very variable class of beings. Heck, one notable Exarch of Kord is Krag-Ik Eight Eyes... an incredibly powerful Beholder who decided to give up on his usual evil and become a worshipper of the God of Storms and Battles.
In Warhammer 40000, Exarchs are the leaders of the Eldar Aspect Shrines on the Craftworlds and lead the warriors of Khaine both in the worship of their god and on the battle field as warriors without peer.
These are individuals who have become so obsessed with their profession, or find they like the emotional highs of killing and maiming too much, that they become incapable of separating their protective War Mask, the part of their psyche separating their warrior selves from their everyday selves, from their core personality. After this, they become stuck on one of the aspect warrior paths and are thereafter referred to as Exarchs.
These Eldar have become the greatest experts of a particular kind of warfare on their Craftworld and are given powerful weapons and armor usually kept within the temples. They master many skills through long experience, which they then pass on to other Eldar who take up position for a time in the temple. Exarchs act as priests of Khaine as much as warriors, for they are the chosen holy warriors of the bloody handed god.
On the battlefield they can really wreck your shit, as they can wield wargear and skills that make their squads as a whole more useful and deadly. Of course, this is countered by the fact that they cost an arm and a leg in points, but if used correctly, they will mow down the competition.
They have another, less pleasant duty. When the need is dire and the Avatar of Khaine must be awoken to help the Craftworld, it requires the sacrifice of an Exarch anointed the young king, who will help the god to manifest for a time. The Exarch chosen as the young king makes the ultimate sacrifice: giving their body and soul to allow their burning god to step forth once more to grind the enemies of the Eldar underfoot.
Many exarchs tend to die in some horrible/vain/glorious way in the fluff, just to make whoever it is they're fighting seem even more badass, a distinction they share with the Avatar and Terminator sergeants.
All Craftworld Eldar practice the Paths, forms of mental concentration meant to focus their immense emotions into a single task, thus preventing themselves from being overwhelmed by sensation and falling prey to Slaanesh, the god born from the ancient Eldar's excesses and rampant kinkiness. There is a danger though. An Eldar may become obsessed with achieving perfection along a path and become fixated on it, unable to ever follow another path again. This is referred to as being lost on the path, and the Eldar is treated with a strange sense of loss by other Eldar, though many masters of the different paths are also widely respected.
The difference with Exarchs is that while they are respected by the rest of Eldar society, they are simultaneously feared and reviled. When not actively participating in warfare, or training students at their aspect shrine, it is extremely rare to see them out and about in everyday Eldar society. Furthermore, those lost on the Path of Khaine are not permitted to join the Infinity Circuit when they die, since their agitated spirits would unbalance it. (Note: this contradicts the fluff surrounding Wraithlords, more on that below)
Beginning as just a way to vent pent up emotions, the protagonist: Korlandril takes part in only TWO battles before losing his way and becoming an Exarch, this is noticed by everyone else as a seeming descent into madness since Eldar can empathically pick up on the mental states of others around them.
Korlandril ceases being capable of behaving as anything other than a warrior even when off of the battlefield and makes a disgraceful display at a funeral where people ostracise him for failing to remove his war mask in public and cast him out.
Finding his way to an abandoned aspect shrine (because Craftworld are full of empty spaces, there were spare shrines just lying empty) the shrine reacted to him by providing him an Exarch suit which was a miniature infinity circuit containing the personalities of the previous wearers, so Korlandril becomes a gestalt consciousness named Morlaniath, though there were only a few personalities present and so Korlandril was not lost in the merge and so could still be "reached" and influenced on a personal level by people who knew him. This is actually described in Path of the Seer where his friend Thirianna becomes a seer and follows the thread of his fate, noticing it joined the thread of others when he became an Exarch.
Later down the line, Morlaniath witnesses the destruction of Karandras during a battle on the craftworld, but when coming into contact with the fallen phoenix, the personalities of Morlaniath merged to join Karandras and repaired the Phoenix Lord who appeared to consist only of energy, which was likened to a galaxy of souls all operating under the single overriding instinct of "Karandras". At this point Thirianna realises that her friend is now lost.
This appears to be the end-fate for all Exarchs eventually... that their souls will eventually coalesce and join one another outside/separate of the Infinity Circuit, adding layers and layers of personality with each suit bearer. The only fundamental distinction between Exarch and Phoenix Lord (other than physical substance) seems to be that the Phoenix Lords are just extremely OLD Exarchs consisting of THOUSANDS of souls.
Exarchs and Wraithlords
As mentioned earlier, since Exarchs do not join the infinity circuit when they die, they merely await the next person to join them and add another layer of personality, which causes a conflict from some sources which indicate that Wraithlord "souls" come from the Infinity Circuit. Since Spritseers are using Exarch "souls" to empower their wraith constructs, then they would have probably stolen the spirit stone from a broken suit or from an empty aspect shrine somewhere rather than drawing the soul directly out of the infinity circuit like they would do with Wraithguard or Wraithblade constructs.
From the book "Valedor" it is clear that Wraithlords are indeed Exarchs, and act as an alternative set of armor for the spirit stones that constitute the gestalt consciousness of the Exarch. If for whatever reason that the armor isn't used on a living Eldar (for instance: no bloodthirsty Eldar to take up the mantle of "Exarch" or the suit gets lost or damaged before the spirit stones were recovered) the Wraithlord shell makes for a pretty good host for the Exarch that never rests and is constantly looking for war and murder. Though this could be seen to be a disgustingly desperate use of resources, since although Wraithlords are comparatively powerful, that gestalt consciousness can no longer be added to by successive resurrections of personality, nor will that Exarch ever return to their shrine and be able to train another generation of Aspect Warriors.