Whenever a 'Mech falls, it suffers damage and its warrior may suffer damage. The amount of damage taken by the 'Mech depends on its weight and the distance it falls. Whether or not the MechWarrior suffers an injury depends on a Piloting Skill Roll.
Other types of units do not fall. If they are moved out of their hexes for any reason, they are considered displaced and use the Unit Displacement rules.

Facing after a fall When a 'Mech falls, it takes damage and its facing may change.
This facing change determines the 'Mech Hit Location Table used when assigning damage from the fall. To determine the unit’s facing after the fall and the area of the 'Mech that takes damage, roll 1D6 and consult the Facing After a Fall Table.
A fallen 'Mech lies prone and face down. 'Mechs that fall on their sides or back automatically roll over to lie face down after taking damage. Any damage to the prone 'Mech from outside sources such as weapons fire or physical attacks is treated as if the 'Mech was standing facing its current facing.

Falling damage to a 'Mech A 'Mech always takes damage from a fall equal to 1 point for every 10 tons that the 'Mech weighs (rounding up) times the number of levels plus 1 that the 'Mech fell; this is the level of the underlying terrain and not the height of the 'Mech. If it fell into a water hex, use the rules described in Falling Damage in Underwater Movement.
Divide the damage into groupings of 5 points each, assigning any remaining points to one smaller group, and determine a hit location for each grouping. For example, a 'Mech that suffers 33 points of falling damage takes six groupings of 5-point hits and one 3-point hit. To determine the location of the damage, use the appropriate column of the 'Mech Hit Location Table, as specified by the Facing After a Fall Table. If the fall occurs during the Movement Phase, resolve the damage as it happens.

Falling damage to the Mechwarrior To determine if the warrior took damage when the 'Mech fell, the player makes a second Piloting Skill Roll after every fall. First, apply all appropriate modifiers to the target number (including all pre-existing damage modifiers, as well as any current situational modifiers such as taking 20+ points of damage in the current phase, and so on). Then add +1 to the MechWarrior’s Piloting Skill target number for every level above 1 fallen. If the die roll result is equal to or greater than this modified target number, the MechWarrior avoided taking damage. If the result is less, the MechWarrior takes 1 point of damage.

Automatic Damage: The only ways in which automatic damage occurs to a warrior whose 'Mech falls is if the warrior is unconscious at the time of the fall, if the 'Mech is considered immobile (for example it is shut down) or if the modified to-hit number for the Piloting Skill Roll to avoid warrior damage is greater than 12. In those three instances, the warrior automatically takes 1 point of damage. In any other instance, there is always a chance that the Piloting Skill Roll can stop damage to the warrior.