Face-down cards have no properties (name, Attribute, Type, Level/Rank, ATK, DEF, effect) until they are changed to face-up. A face-down card being looked at or revealed is still considered face-down and is returned to its original position once the player(s) have finished viewing it.
There are two distinct battle positions a monster can be in: Attack Position and Defense Position. Monsters on the field (and those that would be Summoned) exclusively and always have a battle position. Attack Position monsters are placed vertically while Defense Position monsters are placed horizontally.
Monsters can only be Normal Summoned in face-up Attack Position, and can only be Normal Set in face-down Defense Position; while "Light of Intervention" is active, monsters can be Normal Summoned in face-up Defense Position. If a monster would be Special Summoned but no position is specified, it may be Special Summoned in face-up Attack or Defense Position, but some card effects can Special Summon monsters in face-down Defense Position. When a monster is Flip Summoned, it is changed from face-down Defense Position to face-up Attack Position.
Monsters can only be in face-down Attack Position due to the effect of "Darkness Approaches".
Positions off the field
While positions usually only apply on the field, some effects give cards positions off the field. For example, "Different Dimension Capsule" banishes a card face-down, and "Pharaoh's Treasure" shuffles itself into the Main Deck face-up. Additionally, Pendulum Monsters on the field that would be sent to the Graveyard instead place themselves face-up in the Extra Deck.