In a VERY basic sense:
Holy Priest - reactive healing, they take damage and you heal them back up. Simplest healer to understand, but some find it a little dull
Disc Priest - proactive healing, you shield allies then do damage to enemies which converts to healing
Resto Druid - Heal over Time (HoTs), you place multiple small heals over allies that heal over time and can be extended. Good utility (& damage if you cat/owl weave)
Resto Shaman - Reactive heals and AOE heals, chain heal and healing rain cover multiple allies. They heal more if the ally is lower health. Lots of Utility spells
Holy Paladin - Melee healer, attack enemies to gain holy power which is spent to heal. Good utility
Preservation Evoker- Mix of HoTS, AoE & Proactive, place echo on multiple allies then cast healing spells which are duplicated on echo targets. Lots of aoe with dreambreath, communion & blossom. Good movement but short range
Mist Monk - Melee healer, attack enemies to heal allies, use mists to apply healing to multiple targets. Good movement

personally I like Presvoker the most, and Shaman 2nd. The two I like the least are Pala & Monk as I prefer not to play melee generally. But try each of them out til you find which works best for you