Guide to playing as an Elementalist

Attributes

Air Magic
Air Magic includes a number of efficient, spammable (little/no recharge time, easy to cast in mass quantaties) single-target nukes, many of which have 25% armor penetration, however it only has six AoE (Area of Effect) nukes, Lightning Touch, Teinai's Wind and Whirlwind, which have low damage and are PBAoE (Point Blank AoE) skills, Chain Lightning, which can only strike 3 targets and causes exhaustion, Invoke Lightning, which can only strike 3 targets and causes exhaustion if the caster is not enchanted, and Arc Lightning, which can strike a second target if the primary target is under the effect of a Water Magic Hex. Air Magic skills can knock down foes, and inflict Blind, Cracked Armor and Weakness. Air Magic includes a number of skills which synergize with Water Magic Hexes. Air Magic includes a number of movement speed buffs, including Windborne Speed, the only Elementalist movement speed buff which can target other allies.

Earth Magic
Earth Magic contains a mixture of offensive and defensive skills. Compared to their closest Fire Magic counterparts, Earth Magic nukes are generally more expensive, however many earth magic nukes cause knock down or inflict conditions in addition to inflicting damage. Earth Magic also includes a number of defensive Ward Spells, as well as a number of self-targeting defensive Enchantment Spells and a few armor ignoring damage spells. Earth Magic includes a number of snare hexes, and Earth Magic spells can cause knock down and inflict Blind and Weakness. A number of Earth Magic spells synergize with knock down and weakness.

Energy Storage
Each rank of Energy Storage increases maximum Energy by 3. Most Energy Storage skills involve Energy management and/or self-healing.

Fire Magic
Fire Magic is the most heavily offense-oriented Elementalist attribute. In addition to offensive skills, the attribute also includes several support skills, including a large number of energy management skills. Several Fire Magic skills inflict or synergize with Burning, and two Fire Magic skills, Meteor and Meteor Shower, can inflict knock down.

Water Magic
Water Magic's nukes generally inflict less damage and take longer to recharge than their Earth Magic or Fire Magic counterparts, however, in addition to inflicting damage, they are generally also snare Hexes. Water Magic also includes a number defensive and movement speed buff Enchantments, snare Hexes and skills which can inflict Blind and cause knock down, as well as two Energy management spells, Water Attunement and Glowing Ice. Water Magic also includes a few powerful spike skills, such as Shatterstone and Vapor Blade.

Equipment

Weapons
Since Elementalists do not rely on autoattack damage to kill, meeting weapon requirements isn't necessary. All casting bonuses (HCT, HSR etc) trigger even when you do not meet the requirement.
The three most popular weapon combinations are a staff, a wand and focus item and a martial weapon and focus item.
Staves can have 20% universal HSR, making them useful for builds which use multiple attributes or PvE spells, staves can have Enchanting upgrades, and can provide more Energy than a wand and focus item combination. Meeting a staff's attribute requirement is only necessary for weapon damage, which is generally unimportant for an elementalist.
A wand and a focus item can provide up to 40% HCT and HSR for a single attribute, allowing spells to be spammed quickly. Meeting a wand's requirement is only necessary for weapon damage, however meeting a focus item's requirement is necessary to receive the full Energy bonus.
Martial weapons can combine an Enchanting upgrade with a +5 Energy or 10% universal HSR inscription, making a one-handed martial weapon and a focus item a viable weapon combination. Meeting a martial weapon's attribute requirement is only necessary for weapon damage.

Armor
Like the weapons, you want to stick with headgear matching your build's main attribute, it's generally a good idea to get headgear for several attributes.
You will need to get better armor later in the game. If you walk around in starter armor, then you will be killed by monsters a lot faster than the rest of your party. See the armorer article and the armor collectors category for details on where to obtain armor upgrades.
They might get tired of resurrecting you, so be consciencious about this.

Understanding the Profession

PvE
In PvE, elementalists mostly focus on Area of Effect (AoE) damage to kill as many enemies as possible. Fire magic is the most effective line for AoE damage, and thus "nukers" are quite common.

Early on, low damage AoE nukes like Firestorm are the only spells available. As better nukes become accessible, each brings more and more utility on top of additional damage.

A few noteworthy "nukes" include:

Searing Flames - the basic deal-damage-to-target spell, Searing Flames works best with groups of nukers
Savannah Heat - every bit as powerful as Searing Flames, this spell will cause tremendous damage over time (DoT) to any targets caught in its radius
Rodgort's Invocation is the epitome of nuking; tremendous damage and a lengthy burning effect, all rather spammable with a fast recharge (if you have good energy management)
Meteor Shower, a Prophecies-only spell, combines knockdown with powerful AoE damage.

PvP
In PvP, Elementalists are mostly a utility class - basically, an elementalist with only damage spells on his bar is ineffective. A few damage skills (preferably with high spike damage or lots of pressure potential) mixed in with more flexible skills (such as condition application, condition removal, wards, or melee buffs) is the way to go.

Some Elementalist builds focus on damage, but still, they don't sacrifice utility. Savannah Heat "nukers" are popular in Heroes' Ascent, because they stack high AoE damage potential with a great run buff.

As is true with most professions in PvP, Elementalists don't bring self-heals.

Playing in a Party

Glyph of Lesser Energy can fuel target-other buffs (such as Great Dwarf Weapon).
Wards can mitigate damage party-wide.
Make sure to always have a resurrection of some sort in your skills.

What to Avoid

Ignoring exhaustion
Low health
Considering Energy Storage a form of energy management. Elementalists need energy management skills just as much as other caster classes.

Play Styles

You have:

The fire main, Nuker
Earth Magic- Use armour increasing spells such as Kinetic Armor with Stoneflesh Aura as so that you present a challenge to warriors, while using powerful spells and combos such as Ebon Hawk followed by Stoning, and if warriors are surrounding you, use aftershock.
Air Magic- Use armor penetrating skills like Lightning Orb in succession on a single target. Make sure you take out the monk first or keep him distracted, as skills like Word of Healing allow them to heal the target just as quickly as you are decimating him.

Choice of Secondary Professions

Monk, Ritualist or Paragon is often used to provide a reusable Resurrection skill.

Ranger: Serpent's Quickness can be used to mitigate the long recharge times of many Elementalist skills.
Monk: Glyph of Sacrifice can be used to quickly cast a resurrection spell.The high amount of energy available because of Energy Storage combined with skills like Glyph of Lesser Energy or Ether Prodigy allows casting largely inefficient heal spells like Heal Party. Monk secondaries also have access to reusable resurrection skills.
Necromancer: The Sunspear skill Necrosis can be used in combination with Searing Flames or other skills which inflict burning for a low energy, spammable direct damage skill.
Mesmer: Arcane Echo and Echo are often used to quickly cast multiple copies of powerful spells such as Meteor Shower. Mantra of Earth in conjunction with Stone Striker and Stoneflesh Aura and/or Armor of Earth can be used to effectively mitigate most if not all damage due to high AL.
Assassin: In PvE, Assassin's Promise can be used to spam energy-efficient but slow-recharging ADOT spells. Shadow Form, in conjuction with Death's Charge and High Damage Fire or Earth spells can be used for Solo farming bosses or even certain areas in the Domain of Anguish.
Ritualist: Death Pact Signet can be used in emergencies to quickly resurrect a key party member, such as a Monk, with high Energy.
Paragon
Dervish: Mystic Regeneration is a powerful heal when combined with several Elementalist enchantments (such as Attunements).