As I said earlier, the Ascendant Council encounter is chaotic and thus is is unsurprisingly complex as well. This is due to the numerous number of boss abilities that appear during the span of the fight. To just survive the fight yourself, you'll need to familiarize yourself with all the mechanics so you can respond accordingly. Check out L2R's guide to Ascendant Council if you need to learn the basics.
Overall, the Ascendant Council isn't terribly hard to heal once your raid knows what's going on, but it will definitely keep you busy during the learning process. Most of the abilities are "forgiving" on normal mode, meaning that the damage is only nearly lethal (as opposed to lethal) and provided a player is topped off, a single mistake won't kill him. Players who repeatedly mess up however, will quickly get picked off.
Phase 1 and 2 In the first two phases, there isn't any best way to heal the encounter because the damage is both random and subject to the skill of your raid. To counter that, it's best to identify the slow learners in your raid and try to compensate for their mistakes by keeping them topped off and buffed with a Power Word: Shield or Renew. Aside from those players, though, just play whack-a-mole with whatever spells match the damage you see. Single-target heals and Circle of Healing are strong during these phases, with few opportunities to use spells like Prayer of Healing unless you've sneakily stuck all the problem players in one group. Keep up Prayer of Mending, as usual. Cooldowns of any sort can be used liberally as long as they're available again for phase 3. If you're healing the tank, you should be able to work in low-cost spells like Heal in between your other single-target spells. Regardless of your role, try to conserve as much mana as you can for phase 3 by not being wasteful. Phase 3 is your chance to top the meters if you're concerned with that.
Heart of Ice and Burning Blood Though there isn't too much finesse required to heal through the first two phases, there is one aspect of phase 1 that you'll want to learn properly: dispelling the two nearly identical debuffs, Heart of Ice and Burning Blood. During phase 1, your raid's tanks and DPS will be engaged in separate groups against Ignacious and Feludius, the fire and water elemental ascendants. While doing this, a fire or ice debuff will occasionally be applied to a member of your raid as a DoT which ticks for increasing amounts of damage over time. These debuffs needs to be dispelled, but before that, the afflicted target should run into the alternative DPS group because players in close proximity to the afflicted player will leech a buff from him that increases their damage on the opposing boss.
So to clarify: If a player attacking Feludius is afflicted with Heart of Ice, then runs to the group DPSing Ignacious before the debuff is dispelled, the group of players attacking Ignacious will receive an increased damage buff against Ignacious. To handle this, think back to how you handled Necrotic Plague on the Lich King encounter in Icecrown Citadel; don't dispel the debuff until the player is in the correct location. Arguably, phase 1 can be done without proper dispelling if your raid DPS is high enough, but if you intend to do the fight on heroic difficulty at some point, I'd advise learning how to handle this aspect of the fight now.
Phase 3 Once all the elemental ascendants combine into Captai-- err, rather the Elementium Monstrosity, the real fun of the fight begins. This phase is a DPS race, and your job as a healer is to try and keep up. Make sure you use Hymn of Hope right before the transition or at the very beginning of the phase so you're as close to max mana as you can be. Shadowfiend can be used during the phase, ideally during Bloodlust so your Shadowfiend can use the extra haste for an additional hit or two.
Damage will be high from the various boss abilities; even though you can try to reduce damage by staying spread, kiting the boss will require constant readjustment on behalf of your raid. Melee in particular will need a little bit of extra attention due to Liquid Ice, but Prayer of Healing is a great answer to that. Try to find a spot in the room where you can either be at the center of the fray and be in range of everyone or on the outskirts where you'll be free to move ahead of the kite path of the boss.
One of the bosses' abilities concerns priests specifically, and that's Gravity Crush. Players will be lifted into the air, focused with damage, and then dropped. Since they'll take fall damage from this, Levitate becomes a very legitimate option -- but you need to keep in mind that this will slow players down from getting back to their jobs and can sometimes land them at bad spots in the room. For that reason, I only use Levitate if a player hasn't received enough healing before he falls and I think he may die without it. You can also use Leap of Faith as an alternative to Levitate to grip players to safety before they fall -- something I like to reserve in case one of the raid's MVPs gets picked up and I want to help him get back to his job faster.
You probably won't need things like Pain Suppression or Guardian Spirit on your tanks, so save them for your best DPS on those 1 or 2% attempts. (An extra 8 seconds could make all the difference!) AoE spells like Holy Word: Sanctuary or Power Word: Barrier are extremely helpful in phase 3 as well, but remember that your tanks and melee will be moving constantly, so it's better to position your AoE healing so it hits as many stationary ranged players as possible.
For Divine Hymn, you'll probably want to set up a rotation with other priests and druids (for Tranquility) in the raid and chain them when things start to get overwhelming for normal healing to handle. If you're holy, though, it will be more cost-effective to heal until you drop dead; then save Divine Hymn for when you take on Spirit of Redemption. (If you're completely out of mana though, you can Kamikaze yourself in an ice patch and bring more meaning to the term "improved death.") If you can make it through this phase, you're winning.
Discipline or holy?
The Ascendant Council is another one of those fights for which discipline and holy are both effective. And honestly, if you raid with two or more priests, having one of each is extremely beneficial, since the raid damage is very random. Discipline's strength at spot healing combined with holy's Circle of Healing are ideal for diversifying the healing response the raid needs.
Discipline priests will have a bit more trouble with mana in this fight than they did before the patch, but putting up shields will still be an effective tool, even when you're running dry (something Blizzard developers seem to have addressed, again, with another Rapture buff on PTR.) Switch to Inner Will at the start of Phase 3 to help cut costs on shielding, and don't forget that you can use Power Infusion to reduce the cost of several shields you might want to put out.
Holy priests do have one advantage over discipline on this fight, and that's using Body and Soul. Increased movement is extremely useful on this fight, not so much for saving yourself from hazard but for getting to a location where you can stop and pump out some heals for a few moments without moving. Especially in phase 3 you'll find yourself constantly needing to keep up with the raid and in those cases Body and Soul is a Naaru send.
More major bosses
If you're wondering why we only covered one boss this week, it's just because both Ascendant Council and Cho'gall are rather complex, and I didn't want to short on the quality on either since they're both legitimately difficult fights. Stay tuned for Cho'gall next week -- and as always, if you have any questions or thoughts, leave them in the comments. Good luck!
You may also be interested in reading Priest healing strategies for early Bastion of Twilight.
Spiritual Guidance has the inside line on pre-raid gear, valor point and raid gear, and healing strategies for bosses such as Atramedes and Chimaeron. Newcomer to the priest class? Look into Discipline Priest 101 and Holy Priest 101.