The 4 (5?) Healers.

All spells have a cast time unless otherwise noted.


Priest
The priest is the archetypal healer.  A jack-of-all-trades.  The Priest actually has two raid viable healing specs.  Here is a rundown of the basic priest abilities.

• Flash Heal - A high throughput, low efficiency heal with a low cast time.
• Greater Heal - A low throughput, high efficiency heal with a high cast time.
• Renew - A heal-over-time (HOT) spell.  Ticks every 3 seconds over 15 seconds. (every 3 over 15).  Instant.
• Power Word: Shield - Places an absorption shield that will absorb a fixed amount of damage.  Instant. Cooldown.
• Prayer of Healing - AOE healing spell that hits party members of the target within 30 yards.
• Binding Heal - Heals the target and the priest.  Good for when the priest gets aggro on a mob.
• Prayer of Mending - A buff that heals the target when the target is struck.  It then travels to a nearby ally.  Has 5 charges.  Has a cooldown.  Instant.
• Dispel Magic:  Dispels a magic effect from the target.
• Abolish Disease:  Removes a disease from the target and removes over time.  (every 5 over 20)
• Mass Dispel:  An AOE Magic Dispel ability.  
• Divine Hymn:  A channeled heal (2 over 8).  That heals the 3 lowest targets for a massive amount.  Cooldown.

Buffs:

• Power Word: Fortitude - Increases the Stamina of the target.
• Divine Spirit - Increases the Spirit of the target.
• Shadow Protection - Increases the Shadow Resistance of the target.

Holy Priest
This is THE classic healer.  Often considered to be the jack-of-all-trades healer.  It has recently gotten a bad rap for not being good enough at any one role.  But I always feel better when I have at least one in my raid, because I know they can adapt to any situation, letting us heal any fight.  The Priest has the most healing spells of any healer:

• Desperate Prayer - Instantly heals the priest.  Cooldown.
• Circle of Healing - AOE heal centered around the target.  Instant.  More Mana efficient than Prayer of Healing, but has a cooldown.
• Lightwell - places a Lightwell on the ground which players can click to receive healing.  Very efficient, but hard to get players to use.  Cooldown.
• Spirit of Redemption - when the priest dies, it will stay around as a spirit for a while that can still cast healing spells for 15 seconds.
• Guardian Spirit - Buffs the target that increases healing done to the target by 40% for 10 seconds.  If they die, they will be instantly resurrected with 50% health.  Cooldown.

Circle of Healing is the iconic Holy Priest spell.  When you get a Holy Priest, you are getting a great AOE healer.  Circle of Healing will take care of most of your AOE healing, while Prayer of Healing and Prayer of Mending will take care of the rest of your AOE healing needs.  Holy Priests aren't the best tank healers, but can still manage if needed.

Discipline Priest
Discipline Priest only became raid viable in Wrath of the Lich King.  Previously, it was the Priest's PVP helaing spec.  The hallmark of Discipline healing is its vastly improved ability to shield targets.  A Discipline Priest's Power Word: Shield reflects damage back at the attacker, restores mana/rage, has no cooldown, and is stronger.  Also, a Discipline Priest's crits will place a shield on the target.  They also have many support abilities.

• Inner Focus - Makes your next spell instant and has an increased chance to crit.  Cooldown.
• Power Infusion - grants the target with 20% extra haste and mana efficiency for 15 seconds. Cooldown.
• Pain Suppression - Reduces damage taken by the target by 40% for 8 seconds.  Cooldown.
• Penance - Launches 3 healing volleys at the target in 2 seconds.  Channeled.  Cooldown.

The Discipline Priest is often considered a very capable single-target healer, because of it's buffed shields and Penance.  However, the Discipline Priest is still a capable raid healer, and shouldn't be ignored in that capacity.

Restoration Druid
The Restoration Druid is classified by its vast array of HOT spells.  They come in all shapes, sizes and uses.  It is also identified by Tree of Life form, which buffs the Druid's ability to heal significantly.  This is a common trend with Druids.  Each tree has forms which make its ability to heal comparable to other healing classes.  Without further ado, here are the core spells:

• Mark of the Wild:  Buffs the target with extra armor, abilities, and resistances.
• Thorns:  Buffs the target with a shield that does damage to any target that strikes them.
• Healing Touch:  A basic long-cast direct healing spell.  
• Rejuvenation:  A basic HOT spell. (3 over 15).  Instant
• Regrowth:  A combination direct healing and HOT spell.  (3 over 21).
• Tranquility:  A massive AOE healing spell to all nearby allies.  (2 over 8)  Long Cooldown.  Channeled.
• Lifebloom:  A (1 over 7) HOT spell.  Stacks up to 3 times.  When the HOT completes, it does a large direct heal and refunds half of the mana cost.  Instant.
• Nourish:  A Direct Healing spell that heals an extra 20% if the target has a HOT on them.  
• Rebirth:  The only in-combat resurrection spell.  Cooldown.
• Abolish Poison:  Like Abolish Disease, but for poisons.
• Remove Curse:  Removes a Curse.

The talented abilities are:

• Nature's Swiftness:  Makes your next healing spell w/ a cast time of <10secs is instant cast.  Best used with Healing Touch
• Swiftmend:  Consumes a Rejuvenation or Regrowth effect on the target to heal for the full amount of the spell.  Direct Heal.  Instant.  Cooldown.  No mana cost.
• Tree of Life Form:  Increases healing received by 6% to all nearby allies.  Other talents increase the amount of healing the Druid does while in Form.
• Wild Growth:  A HOT form of Circle of Healing.  (1 over 7).  The healing is stronger at the beginning than the end.  Cooldown.

Druids are often used for tank healing, because the HOTs will automatically heal the tank after they take damage.  They are effective raid healers too.  Rejuvenation is good for spot healing a raid member who takes a little damage.  Lifebloom can be used in many ways.  A single or double stack of lifebloom can be used to spot heal small damage.  Stacking lifebloom can be used for tank healing where you don't let it fall off, since the HOT heals a large amount.  Also, you can stack lifebloom 3 times, but let the tanks health dip so the bloom at the end will bring them back to full.  It's a complicated spell that needs to be learned.  Swiftmend is useful for burst healing that won't interrupt your mana regeneration.

Restoration Shaman
Restoration Shamans are well known as a very capable AOE healer.  This is because of Chain Heal and Ancestral Awakening.   Earth Shield makes a great addition to a raid when placed on a tank.  The basic Shaman Healing abilities are:

• Lesser Healing Wave:  Fast Healing Spell.
• Healing Wave:  Slower Healing Spell.  More mana efficient, but less time efficient than LHW.
• Chain Heal:  A direct healing spell that chains to two other targets.
• Earthliving Weapon:  Weapon enchant that increases healing and gives the Shaman's healing spells a chance to place a modest HOT on the target.  (3 over 12).
• Healing Stream Totem:  Passively heals all allies for a small amount for 5 minutes.  Can be very powerful in certain fights.  

Talented Abilities are:

• Cleanse Spirit:  Removes a disease, poison, and curse from the target.
• Riptide:  A direct heal/HOT effect (3 over 15).  The next chain heal on that target is boosted by 20%.  Cooldown.
• Earth Shield:  A shield with charges that will heal the recipient when they take damage.  
• Ancestral Awakening:  Causes your LHW & HW crits to also heal the lowest health party member for 30% of the original heal.  Passive.
• Tidal Waves:  Causes Chain Heal and Riptide to buff your next 2 LHWs or HWs.  Healing Wave gets a 30% cast time reduction.  LHW gets a 25% crit boost.  Buff lasts 15 seconds or until used.
• Nature's Swiftness:  Exactly like the Druid spell.  Your next Restoration spell w/ a <10 cast time is instant.  Cooldown
• Tidal Force:  Your next 3 LHW, HW, or CH have an increased crit chance.  The first has +60%, the second +40%, the third +20%.

So the Shaman is very capable at raid healing because Chain Heal and Ancestral Awakening will automatically heal the targets who need the healing. Riptide will boost your chain heals and make your LHW and HW more effective (getting more Ancestral Awakening procs).  Along with keeping Earth Shield on the tank, you have a great healing strategy.  Note the passive/proc'd healing effects that contrast with the Priest's and Paladin's wider variety of spells.
Holy Paladin
Paladins are the highest armored healers, which makes them great for PVP.  Holy Paladins specialize in direct heals and are fantastic tank healers.

• Holy Light:  Long Cast, Time Efficient.  Referred to as a bomb.
• Flash of Light:  Short Cast, Mana Efficient.  Not as much throughput as Holy Light.
• Lay on Hands:  Heals the target for equal to the Paladin's max health.  Cooldown.
• Judgement of Light:  Debuff on an enemy that grants allied attackers healing.
• Cleanse:  Removes a poison, disease, and magic effect.
• Sacred Shield:  Buff that absorbs damage from every attack.  One target at a time.  If Flash of Light is cast on the Shield target, it will place a HOT equal to the FOL's value.  3 over 12
• Divine Protection:  Reduces damage taken by the target by 50% for 12 seconds.  
• Auras:  Various resistance auras and defensive auras.
• Blessing:  Various buffs that can be picked depending on class and situation.

Talented Abilities Highlights:

• Light's Grace: Holy Light reduces the cast time of your next Holy Light by .5 seconds.
• Holy Shock: Instant Cast heal with a cooldown.  Good for moving.  
• Judgements of the Pure: Your judgements will grant you spell haste.
• Infusion of Light:  Your Holy Shock will reduce the cast time of your next Flash of Light by 1.5 seconds and increase the crit chance of your next Holy Light by 20%
• Beacon of Light:  Buff.  All healing spells the Paladin casts will also heal the Beacon of Light.  Allows the Paladin to heal a target of than the tank w/o the tank dying.

Also noteworthy

• Glyph of Holy Light:  Holy Light will also heal 5 allies around the target for 10% of the Holy Light's value.

The Holy Paladin is the quintessential tank healer.  But with Beacon of Light, is allowed to heal other targets w/o sacrificing tank healing.  Many of the Paladin's healing effects are passive, procced, like the Shaman.

Notes

If you made a Holy to Natural spectrum for the lore/design perspective, it would go Paladin->Priest->Shaman->Druid.  You'll notice that the Paladin is mostly Direct Heals, and as you go the healers gain more HOTs, with the Druid being mainly HOT based.  You'll also notice that the Priest and Druid, who have less armor, have more healing spells, and fewer passive healing effects.  The Shaman and the Paladin have more armor, less healing spells, but more healing effects.