Howling Banshees are one of the many Aspect warriors of the Eldar army. They are made up predominantly of females [thanks for setting me straight Nash and Admiral Drax], and are equipped with power weapons and their signature Banshee Mask.
The above models were painted with Bleached Bone, then washed in a watered down Devlan Mud. The hair is Mechrite Red, washed in Devlan Mud, and highlighted with Blood Red.
Banshees have a very specific role on the battlefield, and are one of the three main Elites close-combat choices (the others being Striking Scorpions and Harlequins). Specifically what they are good at are heavily armored foes. This means Space Marines (an even juicier target would be Terminators).
Due to their high initiative and good weapon skill, you can count on the Banshees to always hit first (and kill first) before the opponent gets to respond. They ignore bonuses granted by cover, so they're nice to attack a hunkered down unit and still attack first. The powered weapons is what really sets them apart from the Scorpions, though. While Scorpions (due to their high strength and high number of attacks) excel against horde/mob units with little to no armor, the Banshees do well against heavy armoured elites/troops.
The only process of the Assault Phase that they do NOT excel at is the "to wound" roll. Their Strength is only 3, and this is a weakness of theirs. Everything else they have going for them, and can rip apart a fair number of MEQs (Marine Equivalents). Something to think about if you play Eldar: Have a Farseer nearby to Doom their target. With Doom allowing rerolls to wound, there would be no stopping them. Also, for those of you fighting Eldar ... watch out for Banshees hanging back near a Farseer. Shoot them.
Another tactic I've seen used before involves taking a squad of Dire Avengers with a Shimmer Shield Exarch and Defend. Have your Banshees hang back. Their armor isn't great. Use them as a trump card for when things get hairy mid-game. Use your Dire Avengers to intercept any opponent unit looking to assault a poorly defended unit of yours that should remain shooting. With the Shimmer Shield and Defend, you should be able to hold them nearly indefinitely. Next chance you get, assault them with the Banshees. They should clean up in mostly one round, freeing up both units to repeat the process or attend to separate matters.
A word on Exarchs. You are given three wargear options, and two abilities; Executioner, Mirror Swords, Triskele, Acrobatic, and War Shout.
Executioner: +2 Strength, but it's a two handed weapon. Very useful if you're expecting decent to high toughness in addition to the armor. I had one of these assault a tactical squad of mine (in cover, no less) and saw them nearly wiped out to the man in one round. I decided that table-quarter was suddenly forfeit and gave them a wide berth. I'd suggest you do the same. ;P
Mirror-Swords: +2 Attacks, for when it's not toughness that you're up against, but numbers. This option helps out the Banshees with the numbers game, which is better suited for Scorpions.
Triskele: It counts as a normal Power Weapon AND a Ranged Weapon. 12" range, Strength 3, AP 2, Assault 3. It looks like a great way to get a few extra kills in before the assault even starts! At S3, though, it suffers the same challenge that the other Banshees suffer; the To Wound roll. I plan on making one (most likely same Exarch model, but with the Triskele from Jain Zarr.)
Acrobatics: This grants your Banshees the "Counter Attack" USR. Highly useful, if you happen to be charged in combat, you will still get the +1 attack/model as if you initiated the charge.
War Shout: This ability is in addition to the normal Banshee Mask rules, forcing the opponent to make a Leadership check. If failed, the opponent fights at Weapon Skill 1 for the first turn. This could be very devastating if it succeeds, but in 5th edition it seems that more creatures are getting higher leaderships (especially the types of units you should be aiming Banshees at anyways) so its benefit comes very rarely. Those points might better be spent elsewhere.
Note on Jain Zar: A unit of Banshees that she leads is Fearless. Weapon Skill 7, Strength 4 (6 with the Executioner in close combat) 4 base Attacks, Eternal Warrior, 3 Wounds and a 2+ save. At 190 points, she's more expensive than the Avatar, and it's easy to see why. Oh, forgot to mention she has Furious Charge too. She'll definitely make your opponent sweat, as she's quite the beast in combat. Something else noteworthy, she is equipped with an Executioner AND a Triskele (hers hits at S5 at range). Phoenix Lords always get the best stuff. :P