Damage

damage: amount that may reduce the enemys hit point (hp).; -reduction: see armor

1. Normal, which is reduced by hero's Armor and can be blocked by Shield Block items/skills (Stout Shield, Vanguard...)
2. Magic, which is reduced by hero's Spell Resistance.
3. Mixed - which is reduced both by Spell Resistance and Armor - meaning this is most non-dangerous type of damage
4. Neutral - not reduced by anything - most dangerous type of damage (goes through armor and spell resistance)


Attack Damage

All heroes, creeps and towers deal damage in DotA. The exceptions to this rule are the Animal Courier, the Flying Courier that are the delivery units that exist solely to ferry items, Nerubian Weaver's Watchers and Beastmaster's Hawk.

Heroes, creeps and towers are assigned specific amounts of attack damage, or string "attack 1" in combat section of the World Editor, which is basically the damage that appears in the window detailing the selected unit's statistics.

This "attack 1" is the bread-and-butter physical damage capability of the unit; and more of note, your hero character.

Heroes start off with variable base damage. While the Goblin Techies has a tiny base damage that is hardly better than a ranged creep, the Phantom Lancer has a very significant base damage even at level 1.

It is prudent to have strike a balance between your attack damage and attack speed to maximize DPS. A hero would deal less damage if he attacked for 500 damage every 2 seconds compared to a hero that attacks for 200 damage every half a second.
Ways to boost attack damage

There are 5 ways to augment the damage of your "attack 1":

1. The first way is to level up. Leveling up increases all attributes of your hero, including the primary attribute. Each point you gain in your primary attribute grants a corresponding boost to your attack damage. For example, if you would gain 3 agility points per level, and your hero's primary attribute is agility, you would deal 3 more damage per attack and attack 3% faster.
2. Acquiring items is the primary method to boost damage. Items may either grant a flat-out damage boost, such as a Broadsword that adds 18 damage to your attack, or grant an attribute boost, such as the Boots of Elvenskin, that adds 6 agility. As mentioned above, you gain damage only if your hero's main attribute is agility. Otherwise, you will solely benefit from the other boosts that agility bestows, specifically, a minor increase in armor and attack speed.
3. The next way is to boost damage using spells. Sven's signature God's Strength adds a whopping 150% damage boost! It is important to note that most percentage based attack damage buffs grant the boost based on your hero's base damage plus total number of points in your primary attribute, item bonuses inclusive. Buying items that grant raw damage do NOT boost percentage based attack damage buffs. Acquiring the Double Damage Rune provides a 100% attack damage buff to your hero.
4. Another way is to lower the armor of your enemy. The Stygian Desolator is a deadly item that adds 60 raw damage to your attack as well as inflicting an armor reduction debuff on your target. To gain an insight into damage versus armor calculations, see armor.
5. The final way is to use spells that increase the damage an opponent takes. The most potent damage amplification spells are the Holy Knight's Penitence skill, which increases the damage an opponent takes by 28%, and the item Orchid Malevolence's active skill Soul Burn, which increases the damage an opponent takes by 15%.

Spell Damage

The other way to do damage is to cast a spell that deals damage to your target(s), or "nuke". All heroes have an innate 25% magic reduction. If your spell Shock Wave is at level 4, it will only deal 225 actual damage to an enemy hero instead of the advertized 300. Note that a good portion of spells can do damage to more than one target in its radius - Shockwave deals damage in a line.

Only heroes and powerful creeps can cast spells that do damage. Just like physical attacks, the damage of spells is variable as well. While a level 1 Carrion Swarm will do a measly 75 damage, powerful ultimates such as Laguna Blade can deal up to 950 damage, unboosted. Certain spells do damage based on the percentage of your opponents' hit points, which means that the spell's damage increases with the target's hit points.

There is only one way to increase the potential damage of your spells - level them up! The higher the level, the more damage your spell will do.

Spell Damage is by default Attack Type Spells, Damage Type Magic.

Aghanim's Scepter is a powerful item that can further boost the ultimates of certain heroes in the game, upgrading each level of the ultimate of those heroes.

Items that provide spell resistance lower the damage of nukes cast on you. Alternatively, the Black King Bar can provide spell immunity when activated, or the Linken's Sphere can negate the damage of most nukes. Some spells can do the reverse; that is, make the target more vulnerable to spell damage, such as Decrepify.

A small number of spells can inflict divine damage, or Pure damage that cannot be reduced in any shape or form. A prominent example is the Tinker's Laser, which deals precise damage no matter the spell reduction of the enemy - unless the enemy has been Decrepified.
Conclusion

In conclusion, damage is a core component of the game and must be maximized with limited resources to secure victory. A hero that deals mainly physical damage should consider his item choices carefully. If your hero is more adept at dealing magical damage, your item choices will naturally deviate from the former. Correct placement and position of spells makes or breaks a caster.