Invulnerability prevents the user from taking any damage from basic attacks, specials, super moves, or enemy passives. The effect, such as those provided by The Batmobile, Green Lantern/John Stewart's passive, and the Astro-Harness gear will also protect the user from all forms of status effects, except for damage over time effects that are applied before the invulnerability, which is not cleansed and would resume affecting them after the invulnerability wears off, if the duration permits (the DOT continues to run during the invulnerability, it simply does no damage).
Raven/Prime's health swap is unaffected by invulnerability.
Area Effect behave differently depending on source. Character-inherent Area Effect (such as those from Darkseid/Apokolips, The Arkham Knight, Reverse Flash) is unaffected (primary target still takes no damage, but teammates will take damage as normal), while those from gear (e.g. Fourth World Godly Mace, League Of Assassins Adept Knives) will deal no damage. An exception is Harley Quinn/Suicide Squad's C4 bomb (when teamed with Deadshot), which deals no Area Effect damage against invulnerability.
Passive-based disable special will ignore invulnerability, but gear-based disables need to hit and will have no effect.