Face-Up
Face-up refers to cards that have their face visible (i.e. are "public") on the Battlefield; in opposition to Face-Down cards that are hidden.
Face-up cards can have attributes other than their type (for example, they might have stat values, effects, and other characteristics). They can also hold counters (damage or mana counters), as long as they remain on the field.
Face-up cards on the field cannot normally be put back face-down; in fact as per the Card Lifecycle, they will remain face-up on the field until they are destroyed, point at which they are sent to the Graveyard.
The concept of Face-Up and Face-Down cards is very similar to the same mechanic on Yu-Gi-Oh!.