Like every class, Priests have three talent trees, with each tree focused on a different aspect of being a priest. The Holy tree focuses on the healing side, the Shadow tree focuses on DPS, and the Discipline tree focuses on PVP – specifically things such as resistance to interrupt, silence and stun effects that can prevent you from casting your spells during PvP combat.
Discipline
The talents in this tree tend to focus on effects that are useful in PvP – resistance to crowd control affects such as fears, stuns and improving your instant cast spells and combat spells, as these are the most easily used spells in PvP, as they bypass interrupt effects like a rogue's Kick ability. The early abilities are also useful in Shadow and Holy builds, because early talent options allow for better mana regeneration in combat, an improved Power Word: Fortitude buff, and a more durable Power Word: Shield.

A sample Discipline build might look like this: http://www.wowhead.com/?talent=bxMGzhxtbtEoZxMGd
Discipline (46 points)
5/5 Unbreakable Will
2/2 Improved Power Word: Fortitude
3/3 Improved Power Word: Shield
2/2 Martyrdom
1/1 Inner Focus
3/3 Meditation
5/5 Mental Agility
5/5 Mental Strength
1/1 Divine Spirit
2/2 Improved Divine Spirit
5/5 Force of Will
1/1 Power Infusion
5/5 Reflective Shield
5/5 Enlightenment
1/1 Pain Suppression
Holy (0 points)
None
Shadow (15 points)
5/5 Blackout
2/2 Improved Shadow Word: Pain
3/5 Shadow Focus
2/2 Improved Psychic Scream
2/5 Improved Mind Blast
1/1 Mind Flay
This provides access to the useful Mind Flay spell, as well as increased mana regeneration. Damage is reduced while Pain Suppression is activated, and your Holy Shield will also reflect damage back into whoever is attacking you. Finally, your spell damage is increased by the Improved Divine Spirit effect, as well as when you activate Power Infusion on yourself. Alternatively, you can cast this on a nearby mage or warlock to increase their damage output as well.
Holy
Holy has two different builds that are very common – one that focuses on reaching Improved Divine Spirit for the boost it provides casters in your party or raid, but at the expense of giving up Circle of Healing, the final spell at the end of the Holy Tree. This spell is very useful to have in raiding, as it allows you to heal an entire party with one instant spell. If your melee DPS is grouped into one party, this can be very handy as it allows you to heal everyone who might be taking AoE damage from an enemy. The two builds for Holy tend to look like this:
Improved Divine Spirit
http://www.wowhead.com/?talent=bxRhzhxzbZfxt0fMqVzc

Discipline (23 points)
5/5 Unbreakable Will
1/5 Silent Resolve
2/2 Improved Power Word: Fortitude
3/3 Improved Power Word: Shield
1/1 Inner Focus
3/3 Meditation
5/5 Mental Agility
1/1 Divine Spirit
2/2 Improved Divine Spirit
Holy (38 points)
2/2 Healing Focus
3/3 Improved Renew
5/5 Holy Specialization
5/5 Divine Fury
1/1 Holy Nova
2/2 Holy Reach
3/3 Improved Healing
2/2 Healing Prayers
1/1 Spirit of Redemption
5/5 Spiritual Guidance
5/5 Spiritual Healing
1/1 Lightwell
3/5 Empowered Healing
Shadow (0 points)
None
Circle of Healing
http://www.wowhead.com/?talent=bxRhzhbZfxtcfMqVzVo
Discipline (17 points)
5/5 Unbreakable Will
1/5 Silent Resolve
2/2 Improved Power Word: Fortitude
3/3 Improved Power Word: Shield
1/1 Inner Focus
3/3 Meditation
2/5 Mental Agility
Holy (44 points)
2/2 Healing Focus
3/3 Improved Renew
5/5 Holy Specialization
5/5 Divine Fury
1/1 Holy Nova
3/3 Inspiration
2/2 Holy Reach
3/3 Improved Healing
2/2 Healing Prayers
1/1 Spirit of Redemption
5/5 Spiritual Guidance
5/5 Spiritual Healing
1/1 Lightwell
5/5 Empowered Healing
1/1 Circle of Healing

Shadow (0 points)
None
The main difference between the two is the build for Circle of Healing pulls more points out of the Discipline tree to instead finish out Empowered Healing, and fill out Inspiration. The ability to increase the tank's armor by 25% with your critical heals can help the tank a great deal in reducing the amount of damage he is taking during boss fights where the boss deals a great deal of physical damage.
Neither build puts a great deal of points into Silent Resolve in the Discipline tree – generally, you shouldn't have a lot of trouble with aggro if the tank is doing a good job. If enemies are running loose and you're having to cast large heals on anyone but the tank, chances are good that you'll end up being attacked with or without this talent. None the less, if you find yourself consistently grouping with tanks who have trouble holding aggro against you, you can remove the 3 points from Inspiration and one point from Mental Agility in the Holy Circle spec to fill out Silent Resolve. In the Improved Divine Spirit build, you can take any combination of points from Mental Agility or Empowered Healing to fill out Silent Resolve.

Shadow
The Shadow talent tree is where you tend to do the most damage with the least mana expended as a priest. In the Shadow tree, you'll end up spending most of your points into it, unlike the other sample builds where 17 or more points go to another tree.
http://www.wowhead.com/?talent=bxMhZZxGMtrtRhtEo
Discipline (10 points)
5/5 Unbreakable Will
2/2 Improved Power Word: Fortitude
3/3 Improved Power Word: Shield
Holy (0 points)
None
Shadow (51 points)
5/5 Spirit Tap
3/3 Shadow Affinity
2/2 Improved Shadow Word: Pain
2/2 Improved Psychic Scream
5/5 Improved Mind Blast
1/1 Mind Flay
2/2 Improved Fade
2/2 Shadow Reach
5/5 Shadow Weaving
1/1 Silence
1/1 Vampiric Embrace
2/2 Improved Vampiric Embrace
3/3 Focused Mind
5/5 Darkness
1/1 Shadowform
5/5 Shadow Power
5/5 Misery
1/1 Vampiric Touch
With this build, you gain access to the more durable Power Word: Shield and your Vampiric Embrace ability heals you and your party for 25% of the damage you do. You also gain access to Silence, allowing you to stop enemy spellcasters temporarily, and many talents to increase the damage of your Mind Blast and Mind Flay abilities.