Exarch
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 armour usually kept within the temples and master skills through long experience, which they pass onto 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 up as they can possess 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 used rightly 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. So the Exarch chosen as the young king makes the ultimate sacrifice, as they give themselves 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.
Background
All Craftworld Eldar practice the Paths, forms of mental concentration to focus their immense emotions into a task so they are not overwhelmed by sensation like their ancestors were and fall 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. They become referred to being lost on the path and are treated with a strange sense of loss from other Eldar for this, although many masters of the different paths are also widely respected.
The difference with Exarchs though 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)
The first book in Gav Thorpe's Path of the Eldar series details one individual's journey all the way from joining the Path of the Warrior all the way up to becoming the Phoenix Lord Karandras.
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.
This creates the problem with the description of Wraithlords 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.
If Spritseers are using Exarch "souls" to empower their wraith constructs, then they would have probably stolen the soul stone from a broken suit or from an empty aspect shrine somewhere rather than drawing the soul directly out of the infinity circuit, which seems to be a disgustingly desperate use of resources, since although Wraithlords are comparatively powerful, that gestalt consciousness could have been re-used later to create another Exarch and train another generation of aspect warriors.
Either that or it has been misrepresented as "Exarch" when in fact it might have originally meant "Aspect Warrior" who can still be warriors of legend, just not lost on the path.
Alternate view: In 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). For whatever reason that the armor isn't used on a living eldar (no bloodthirsty eldar to take up the mantle or the armor is somehow lost while the spirit stones were recovered), the wraithlord shell is a pretty good host.