Increased Damage

As you probably are aware, Damage is very important in Diablo 3 unless of course you are healer and your only job is to heal. If you are any other type of character and especially if you play solo, increasing the amount of damage you can deal will equate to how quickly you can take down enemies. After Nightmare difficulty, killing enemies before they kill you becomes your main concern.

Increased Damage can be achieved by buying, trading for, crafting or finding a high DPS (Damage Per Second) weapon for yourself.  Other types of gear and enchantments can also increase your damage.

Also See:
Increase Critical Hit Damage

In addition to this, every Class in Diablo 3 has Skills that increase damage. It will be important to utilize these, in order to make the most powerful character you can.

The Following Is A List Of Each Class' Skills That Can Increase Your Damage:

Barbarian Skills That Increase Damage

Bash (Punish)
Battle Rage
Battle Rage (Marauder's Rage)
Wrath of the Berserker (Insanity)
Weapons Master
Berserker Rage

Demon Hunter Skills That Increase Damage

Marked for Death
Steady Aim (Passive Skill)
Cull the Weak (Passive Skill)
Fundamentals (Passive Skill)
Archery (Passive Skill)
Ballistics (Passive Skill)

Monk Skills That Increase Damage

Deadly Reach (Foresight)
Breath of Heaven (Blazing Wrath)
Crippling Wave (Breaking Wave)
Mantra of Conviction
Combination Strike (Passive Skill)
Guiding Light (Passive Skill)

Witch Doctor Skills That Increase Damage

Hex
Sacrifice (Provoke the Pack)
Mass Confusion (Paranoia)
Big Bad Voodoo (Slam Dance)
Fetish Army (Legion of Daggers)
Pierce the Veil (Passive Skill)

Wizard Skills That Increase Damage

Slow Time (Time Warp)
Arcane Torrent (Disruption)
Familiar (Sparkflint)
Glass Cannon (Passive Skill)

Frost Nova (Bone Chill)- Enemies take X% more damage while frozen or chilled by Frost Nova.

Black Ice (Passive Skill)- Cold damage dealt to chilled and frozen targets is increased by X%.

Conflagration (Passive Skill)- Dealing Fire damage to enemies applies a burning effect, increasing all damage done to them by X% for X seconds.

Arcane Dynamo (Passive Skill)- When you deal damage with a Signature spell you gain a Flash of Insight. After 8 Flashes of Insight, your next non-Signature spell deals 50% additional damage.

Follower Skills That Increase Damage

Scoundrel (Hysteria)
Enchantress (Scorched Earth)