Casual Mode (Newcomer Mode in the PAL version of Fire Emblem Awakening) is a play mode which was introduced in Fire Emblem: New Mystery of the Emblem and has been present in its successors.
The Fire Emblem series is known for its permanent death system. When a Player Unit's health reaches zero, they die (or permanently retire, depending on their importance to the story) and can no longer be used in any future battles unless revived via certain Staves or by restarting a level. Casual mode disables this feature, meaning that if a Player Unit's health reaches zero, they will retreat from the battle, keeping all experience and items gained during the map. They will be able to be used in the future after the chapter or battle ends.
Casual Mode does not affect Lord or Avatar characters, so if their health drops to zero, it still results in a Game Over, with the exception of Fates and Three Houses. It also does not affect NPC and Ally characters.
In Fire Emblem: Three Houses, there are two battles that have casual mode on, even when the player selected Classic Mode, that being the Mock Battle in Chapter 1, and the Battle of the Eagle and Lion in Chapter 7. This is because in both cases a mock battle is being held, so there is no real harm being done to the player's army.