No other class has so many tools for healing at its disposal as the Priest. This makes him a very flexible healer able to perform almost any healing duty. While he may not be the strongest healer on any of the healing fields, the flexibility makes up for that.
Prayer of Healing is the main tool for group- and raid-healing, as it can target any group from the raid. A Holy Priest will have the instant Circle of Healing to supplement his group-healing capabilities. Discipline Priests are able to shield whole groups with Power Word: Barrier. Holy Nova can be glyphed to heal for almost as much as circle of Healing does, but it is limited to the own group and has a small radius. Then tehre is Prayer of Mending, a healing spell that will jump to different targets and heal them whenever they take damage.
For single target healing the Priest has Flash Heal, Greater Heal and Renew, a heal-over-time effect. Power Word: Shield is able to absorb smaller amounts of damage. Discipline Priests additionally have Penance, a channeled healing spell. Desperate Prayer is a self only instant heal with a very low mana cost. Apart from that a Holy Priest is able to lay down a Lightwell where group and raid members are able to receive a powerful hot whenever they need it. While Shadow Priests are unable to cast healing spells in Shadowform they get a special way of healing: Vampiric Embrace will heal the Priest and his group for a percentage of the damage he deals.