Strengths & Weaknesses
Priests are one of the strongest classes in 1v1 PvP and especially in group PvP. With the recent talent revision, there are multiple viable PvP specs, both for healing and doing damage. While paladins are probably the best PvP healers due to their survivability, the combination of quick heals and shields make priests amazing. There are two major types of PvP priests, with a variety of different builds for each.
Shadow Priest
The PvP shadow priest is a force to be reckoned with 1v1 or in a group supported by another healer. The main source of power of a shadow priest comes from Shadow Word: Pain, which is instant cast and mana efficient. The added bonus of this spell is that it benefits fully from +dmg gear modifiers, meaning that it can dish out huge damage. With good gear and trinkets, 400 damage a tick is easy to achieve, and can conceivable go much higher.
Priests also have the best full Silence in the game. It lasts for 5 seconds, but doesn’t have a counterspell element like the mage’s version. Proper knowledge of other classes is essential to knowing when to use Silence. Interrupt heals on any healing class. When facing damage classes, first work out what school of magic they are specced in (for example, Frost mages will have Ice Barrier active, while Conflag warlocks will cast immolate on you, and probably won’t have instant-cast corruption), and time it to interrupt one of their major spells. Good players will attempt to get you to blow your Silence early, since it is on a cooldown. Try not to fall for that!
Another key spell to use is Vampiric Embrace. It heals for a good amount, especially when you have a few sources of damage ticking on the target. When combined with Shadowform and some good armour (plus Inner Fire), a shadow priest can be extremely difficult to bring down.

Most of all, don’t be afraid to drop out of Shadowform to heal your team when they need it. If you take damage and you are on your own, bandage up, but if you are supporting the flag runner, for example, don’t stay in Shadowform. Getting number one on killing blows means nothing if your teams loses 3 caps to nil.
Support Priest
A support-focused priest will be more effective and can definitely be more annoying in the long run that a DpS priest. The other team will immediately target you on every skirmish (for good reason), and you see results from playing this role.
The key to playing a good support priest is to find yourself a good team-mate. The best choice for heavy-holy priests (with a smattering of Discipline) is a warrior, because they can deal out huge damage and will often be focused on by the enemy. If you are heavy in Discipline, with Holy support, you should team up with a mage, for huge numbers with Power Infusion (which you should use as much as possible, as it has such a short cooldown).
Always keep Renew and a shield on your target. You may want to save Power Word: Shield for when your partner is affected by Mortal Strike (a warrior ability which reduces healing effects by 50%, but does not affect shields. Most PvP warriors have this ability). You HAVE to have Healing Focus to be a PvP healer, but since it’s on Tier one, most every priest has it anyway. Use Flash Heal rather than Greater Heal, because PvP damage is very spikey, and often 1 extra second can mean death.

Throw up Pain whenever you can, as even in full healing gear it adds at least a bit of damage. You can use Smite, Mind Blast and Holy Fire when you get a chance to help add some extra damage. Sometimes it’s fun to use Power Infusion and a damage trinket, and run into a group of opposition players and spam Holy Nova, seeing how long you can last (as long as they don’t stun you, you would be surprised at the survivability this allows).

Engineering can help make up for your lack of damage, through grenades or the Death Ray. If you don’t know anyone in your battleground or world PvP area, just watch and heal for a while, and if you find a competent partner whisper them and group together. They will appreciate the heals, and you will benefit from the damage they can put out.
I find that playing support is particularly fun and effective in Warsong Gulch, where you can heal the flag carrier to frustrate the enemy, dispel any magic-based slows, and fear the chasers. Most players will always focus on the flag carrier and ignore you, which usually ends in failure for them.
General Strategies
Psychic Scream is an extremely powerful ability, but should not be a crutch. This is particularly true for the Alliance, because of the huge proportion of Undead players, who can remove your fear every 2 minutes. Many classes have easy ways of removing it as well. Learning to play without having to rely on fear will benefit you greatly in the long run. As for avoiding Psychic Scream, a lot of the time it is easy to do. It only has an 8yd maximum range, so if you see a priest running towards you (and most likely a number of your teammates), make sure you stay out of range. If you are a mage, blink.
Always dispel your target (this goes for any priest). Dispel now checks for debuffs, so you can’t ever waste mana by using it. Additionally, you should dispel magic on yourself if it is particularly harmful (but be warned, some smart players will use low-rank spells and DoT’s to get you to waste some mana on dispelling), and any players on your team that you feel would need it.

Priests (like any caster) rely on mana to do any significant damage, but even more so because they have no inherent ability to return their mana (like mages through gems/Evocation, and warlocks with Life Tap). For this reason, it is important to conserve the limited resources you have. This means not spamming your DoT when a class with Dispel Magic is around (other priests, paladins, and warlocks with a felhunter out). This may also mean wanding when necessary. A good damage wand is highly desirable, particularly for a healing priest, as you will want to conserve mana for heals, and wands can put out good damage.
Most of all: Heal! Too often priests refuse or forget to heal themselves and others, which is the real strength of the class. Smashing others with a shadow spec is great, but most of the time it won’t help you win a battleground. Healing does.
Macros
Smart Targeting Heals
This checks nearby friendly players and casts Flash Heal on them if they are below 70%.
/script for i=1,40 do TargetNearestFriend(); if UnitHealth("target")/UnitHealthMax("target") < 0.7 then if UnitIsPlayer("target") then CastSpellByName("Flash Heal") end end end; TargetLastEnemy()
Self Casting Without Losing Your Target
This will cast Flash Heal on yourself without losing your target. You can substitute Flash Heal for any other heal you want.
/script CastSpellByName(‘Flash Heal’, 1)
Arena
In the arena, priests are most commonly used as healers and occasionally as damage dealers. As healers, they are excellent, if specced correctly and with quality gear.
Priest healers need Blessed Resilience to survive. Pain Suppression, while good on paper, is too easily dispelled, and is limited in use. Blessed Resilience, on the other hand, is a passive talent and thus is constantly applied.

Priests need lots of stamina, resilience, and +healing to be effective in arenas. This will help in their survivability. Psychic Scream, with its long cooldown and numerous counters, is not an effective way to prevent damage – the major way is to just live through the damage with Shields and Renews.
Shadow Priests can be used for DpS, and can do quite well. However, they are often targeted early, and although have good damage reduction with Shadowform, they are still quite easy to kill. However, Shadow Word: Pain and Death can do excellent damage and the healing from Vampiric Embrace can be excellent for keeping your group topped off.