Effect Type
All actions that affect a unit are associated with a damage type. Damage Type determines whether Armor or Magic Armor mitigates the damage and whether it's blocked by Magic Immunity or Physical Immunity, if at all. Damage types also apply to targeting, and different properties of an ability can have different damage types.
Physical
Physical damage is mitigated by Armor, and is blocked by Physical Immunity. All auto-attacks deal Physical damage (with the exception of Harkon's Blade), as well as select abilities.
Superior Physical
Superior Physical ignores Physical Immunity. It is a mostly unused damage type, and in-game it is used with Mixed damage. Currently, there are no abilities that exclusively possess the Superior Physical damage type.
Magic
Magic damage is mitigated by Magic Armor, and is blocked by Magic Immunity. Magic damage and type is mainly associated with abilities.
Superior Magic
Superior Magic ignores Magic Immunity. Superior Magic is usually used for status effects, providing a way to disable a Magic Immune hero, and most forms of Superior Magic are used in ultimates.
True
True damage is not mitigated by Armor or Magic Armor and will ignore Magic Immunity and Physical Immunity. It will still be reduced or amplified by forms of regular reduction/amplification. While it will always deal the exact amount of damage listed, it won't deal any extra damage the target has negative armor.
Mixed
Mixed damage is mitigated by both Armor and Magic Armor separately, for example. If an attack deals 400 mixed damage to a target with 25% physical damage reduction and 25% magic damage reduction, it will deal 200x0.75 + 200x0.75 = 300 damage. Mixed damage is blocked by both Magic Immunity and Physical Immunity. Mixed damage is usually associated with abilities that deal both physical and magical damage in one instance.