For most people (I will admit, me included) the first word you think of when the word Priest is mentioned is heals. Since the beginning of Wow, priests have been the premium healing class in the game. That will always be what they are known for, but blizzard has done a good job evolving them into more solo-friendly characters.
Strengths: Depends on spec. Priests obviously have solid self-healing abilities and with shield and inner fire (do yourself a favor if you're soloing, macro a self-only shield), they have a decent amount of durability relative to other casters. If specced into shadow, they can deal impressive single target damage with little downtime. Shadow priests function very similarly to affliction warlocks.
Weaknesses: Multiple mobs are a major problem. Priests lack any real AOE damage and they don't have the variety of DoTs to handle groups like warlocks. Also, like all casters, they are inherently squishy and since most priests will stack +int, +spirit and +spellpower, this weakness is magnified.
Strengths: Depends on spec. Priests generally are either caster DPS or Healing. Shadow priests bring a lot to groups from a mana regeneration standpoint. If specc'd into Holy they offer extemely solid single target and group heals.
Weaknesses: While priests in both shadow and Discipline are capadble of healing, a lot of players (myself included) refuse to run with "off spec" healers. This is not as much from a mechanical lack of healing ability (unless they're a total n00b and try to heal in shadowform), but more an experience standpoint. Generally, players without experience healing lack skill in it, so having abilities to heal with no skill in their use is almost as bad as not having abilities. This is not to say shadow priests are incapable of healing, but from an armory check, how can someone know?
Strengths: Depends on their spec. Holy and Discpline priests should avoid doing anything other than healing in PvP. Shadow priests can be difficult for casters and melee alike due to solid single target DPS and mana drain abilities. Also priests have both mind control and fear which can be useful tools in PvP.
Weaknesses: See above. Shadow priests have tools for PvP, other specs really don't.
Holy: If you want to heal instances, holy is the way to go. It is what it says it is, heavy on PvE healing, not so great at anything else. Although priests do get a number of cool Holy damage spells, this line doesn't do much for them so it will be harder to level with than the other 2.
Discipline: Discipline is a good middle ground between Holy and Shadow. It gives a good balance between soloing and healing. Discipline mainly focuses on increasing stats and efficiency. This is also a very commong secondary spec for remaining talent points.
Shadow: Shadow is the main choice for soloing and caster DPS focusing priests. As I said previously, shadow is very similar to affliction warlocks. Shadow gives access to more powerful DoTs and mana and health draining along the tree. It is capable of very respectable single target DPS, however it lacks the AOE abilities of the other caster classes.
Most people think of priests as walking healbots, which isnt necessarily the case. All priests can heal to an extent, and if that's your goal, priest is a good class choice. If you are looking for a DPS class with some utility and off-healing abilities, it is harder to find better than shadow priest. However, if you mainly want to DPS, there are other classes that offer a better mix of abilities.
Mid Afternoon Quickie
7 months ago