Phoenix Lord
The Phoenix Lords are the founders of the Eldar's Aspect Warrior shrines, and are the authorities on what it means to practice their respective Aspects. They embody their Aspects so completely that their students often get trapped in that Aspect and become Exarchs.
As of the 41st Millennium, the Phoenix Lords are not properly alive any more; their spirits have integrated themselves with their armor, so if they do fall in battle, they can be resurrected by having someone new don the suit, and they're back in action, though the unfortunate Eldar who dons the suit gets overwritten by the original Phoenix Lord's personality and memories.
On the battlefield, they wield upgraded versions of their Aspect's weapons and have all of the powers available to their Aspect's Exarchs (unsuprising, considering that they taught their Aspect everything).
Supposedly, these Lords will be heavily involved in the final battle against Chaos; given that Phoenix King is one of Asuryan's titles, there may be something to this rumor.
Asurmen
Asurmen, the Hand of Asuryan was the first Phoenix Lord, who founded the Dire Avenger Aspect and the Path of the Warrior. He founded more of his shrines than any of the later Phoenix Lords. Many of his students went on to found Aspects of their own.
Jain Zar
Jain Zar, the Storm of Silence was one of Asurmen's first students and later founded the Howling Banshees Aspect. She still visits all of the shrines to her Aspect every century or so.
Baharroth
Baharroth, the Cry of the Wind founded the jump pack-toting Swooping Hawks aspect. He was the youngest (and most energetic) of Asurmen's first class of students.
Arhra
Arhra, the Father of Scorpions (also called the Fallen Phoenix) founded the Striking Scorpions Aspect. Somewhere along the way, he fell to Chaos, and left the Craftworlds. He is suspected of joining the Dark Eldar as Drazhar, Master of Blades and Hierarch of the Incubi (which he got by killing the previous Hierarch and his bodyguards).
Karandaras
Karandaras, the Shadow Hunter is the current Phoenix Lord of the Striking Scorpions. Arhra's greatest pupil, he took on his master's role after his master fell to Chaos (exactly how he did this is unknown). While he is more loyal to the Craftworlds, he is no less mysterious than his predecessor.
Fuegan
Fuegan, the Burning Lance founded the Fire Dragons Aspect. He was thought lost when Arhra tried to destroy Asurmen's shrine, but he turned up later to fight Chaos on an Exodite world. He is supposed to be the one to call the other Phoenix Lords to the final battle against Chaos, and will be the last of them to die.
Maugan Ra
Maugan Ra, the Harvester of Souls founded the Dark Reapers Aspect. He was the first Phoenix Lord to include ranged weapons as part of his Aspect; his personal weapon is the Maugetar (which is a great name for a musical instrument), which is a beefed-up shuriken cannon with a scythe-like built-in power blade, thus achieving the ultimate fusion of dakka and choppy. His Craftworld, Altansar, got sucked into the Eye of Terror, but during Abaddon's thirteenth Black Crusade, he managed to drag it back out. Maugan Ra also managed to somehow solo a Tyranid Hive Fleet. Which is insane.
Irillyth
Irillyth, the Shade of Twilight founded the Shadow Spectres Aspect. He had a vision that his craftworld, Mymeara, would be destroyed unless he took action to end the threat preemptively. He gathered up a bunch of warriors from Mymeara and went to find them, and they were victorious, but they also never returned, so everyone figured that that was the end of the Shadow Spectres. Later, they found the world that he and his army had invaded, an ice planet colonized by the Imperium of Man in the meantime, and so they raided it and got his armor back, thus reviving him and his Aspect (and explaining why neither he nor his Aspect were represented in the Eldar codices to date).
Irillyth (along with the Shadow Spectres) was created and sculpted by Forge World for Imperial Armour Volume Eleven: The Doom of Mymeara.
Drastanta
Tempest of Starlight, lord of the Shining Spears. Drastanta was really full of himself and like to go on needless solo battles. This backfired horribly during the Fall were because of this it got Arhra killed for the first time. After blowing up the Keeper of Secrets who did it. Drastanta went out to found the Shining Spears and then went into self-exiled leaving his weapon behind. His lance is currently with Craftworld Iyanden.
Living proof that Matt Ward can add a character without canon rape.
Why the Phoenix Lords are Badasses
Well, let's start with Asuryan.
1. He is the start of all this Aspect Warrior crap, so the entire reason the faggots still exist is due to him. Whether this is good or bad is up to you.
2. He is soooooo Badass that not only the eldar worship him, but other races even go "god fuck that guy is Awesome, let's worship the bastard.
3. Ultramarines want to be like Calgar. Calgar wants to be like Grandaddy Smurf. Grandaddy Smurf wants to be like Asuryan.
4. Asuryan would not get stabbed with a poisoned knife by the complete hair-obsessed faggot and then fall into a coma. Asuryan would bitchslap the whore with his AWESOME sword, then let his bitches carry the corpse to that statue and awaken it with the body. Then play poker with the EMPRAH. Cos he's a nice guy.