Talent Trees By Purpose

Enhancement

It's the designers stated intent that Enhancement is the default leveling spec for Shamans. It has little to no down time, and a high damage output. The important spells and abilities are Windfury Weapon, dual wielding, Stormstrike and Lava Lash. The drawback of this spec is that not only damage output is high, but also damage taken due to the close-in melee type of fighting. At higher levels, Maelstrom Weapon allows regular instant heals, which remedies this problem greatly.

Elemental

As the ranged DPS tree, Elemental can also be used for leveling. It's much safer since mobs usually are seriously damaged (or even dead) before they can hurt the Shaman, but has the drawback that it needs rather large amounts of mana, leading to high downtime.

Restoration

For the purpose of leveling or solo PvE, Restoration is a rather poor choice. The tree offers next to no enhancements to damage output, and therefore killing mobs takes a very long time. The extreme survivability granted by a pure resto spec is not really helpful - sure, Resto shamans are in fact able to solo mobs nobody else can kill, but it's not really practical to kill an elite 2 levels higher if it takes 10 minutes or more. Going restoration may be useful when leveling in a group environment (with many instances), but even when healing a lot, a hybrid Elemental/Resto spec is better overall than pure Resto.
Group/Raid PvE

Restoration

Healing was the traditional role for Shamans in all groups or raids. Over the years, the other specs have become viable too, but Restoration is still a very well liked specialization for raiding Shamans. Since 3.0.1, healing Shamans even got an interesting spell rotation (since 2.0 it had been mostly Chain Heal spam). At least one Earth Shield is still wanted by every raid.

Elemental

Elemental is also a wide supporter of groups and raids while also contributing major burst DPS. Other spell casters, especially Fire Mages, Boomkins, Destro Locks, and incredibly enough, Holy Paladins all benefit greatly from your presence. These classes depend on spell crit just like Elemental shaman, and they give it out in total of 8%, 5% from Elemental Oath and 3% from Totem of Wrath. These talents are only available to Elemental Shaman, making us friends among other casters. Elemental Shaman have become a rather large tool that raids depend on for the amazing crits, but also a tool to boost the overall damage of the entire raid, since Totem of Wrath also affects melee crit as well.

Enhancement

Enhancement Shaman are also big supporters in groups and raids. They can throw out the standard totems to boost parties and raids just like the other trees, however, thanks to Enhancing Totems, their totems buff a bit more than other shaman. This talent is in the tier one section of the tree, so the other two trees can grab it and be just as beneficial. The main buff they give is Unleashed Rage, boosting melee DPS a considerable amount. Not as big of a supporter as the other two trees, but still packs a punch in any raid or group.
PvP

Elemental

This tree has great potential to throw off opponents that are both melee, and casters. Elemental Warding will fend off up to 6% damage from all sources. Astral Shift will further reduce damage if placed under one of the conditions. Thunderstorm is an amazing way to fend of melee attackers for enough time to throw in a quick heal. Finally Eye of the Storm (Talent) will help when you are trying to get a cast in when there are multiple people on you. These damage reducing talents along with the high damage output will allow for a rounded PvP Shaman.

Enhancement

This tree can be very fun to mess with people, especially when they are caught off guard. Toughness will reduce both movement impairing effects and a good amount of damage since Enhancement Shaman do not use shields. Static Shock will allow 9 charges of your Lightning Shield, therefore reducing the times you need to refresh it during a fight. Maelstrom Weapon will allow you to toss in some extra damage or a heal at a reduced or no cast time. Finally, Feral Spirit will make you much more intimidating. These wolves are your pets and are affected by Bloodlust. Jumping an enemy by surprise with both wolves, Bloodlust, and some high damage output with Windfury Weapon and both instant strike attacks, many enemies will feel very helpless and intimidated.

Restoration

This tree definitely needs some work in order to fully understand how to use it in smaller PvP situations. In large battlegrounds, you would just heal a DPS to keep them up while running through opponents. However, solo PvP can be slightly less fun as little damage output makes you a vulnerable target to any DPS. Tidal Force will allow your next three heals to have a higher crit chance, allowing bigger heals when you need them. Nature's Swiftness will become your best friend when you have little to no health remaining and need that quick heal. Focused Mind will reduce many silence and interrupt effects that will be used against you. Nature's Guardian doesn't seem like much, but it procs more often then you would think, helping you to toss in another heal before getting killed.Earth Shield will more commonly be placed on you as it will help keep you alive. Finally, Riptide is our instant cast spell that will automatically heal us and place a HoT on our target, increasing survivability a lot more. This tree can be pretty hard to get used to, but once you get the hang of it, no one can stand in your way.