The idea is that the target is frozen solid, unable to move and trapped inside the icy tomb, able to do nothing but watch their enemy approaching with the doubtless intention to shatter the frozen victim into pieces. This does not necessarily mean that the victim is going to die: Anya survived several days of being chained by the freezing curse Nihlathak has put on her, and recovered instantly when the curse was removed. In both games, a target that was frozen, but not dead, continues to fight normally when they thaw (even though their blood vessels should be torn apart by the blood turned ice).
Freeze can only happen as a result of unsaved Cold Damage that is specifically stated to be able to Freeze target (while to Chill the victim, any Cold attack will suffice), or a special effect directly stating that it can Freeze an enemy.
The Frozen target is blue in color and paralyzed completely, and if they die from the freezing attack, or die before the Freeze wears off, they shatter into pieces and leave no corpse. This is especially useful when fighting monsters that can be resurrected, such as Fallen. Freezing enemies is not recommended for Necromancers and for most builds connected to corpses.
The Frozen victim may not attack, move or cast spells. Act Bosses, Super Unique monsters, Champion monsters, and other players cannot be frozen; they will, instead, be Chilled. Some items in game have the Half Freeze Duration or even Cannot Be Frozen traits, and cold resistance reduces duration of any Freeze effect. Some monsters, including Frozen Horrors, are entirely immune to Freeze.
Skills with the Freezing effect are Glacial Spike, Ice Blast, Frozen Armor, Ice Arrow, Freezing Arrow, Phoenix Strike and Blades of Ice. Freeze usually has a shorter length than Chill, but is considered a different effect than Stun.
Chance of freeze / chill is ( 50 + (AttackerLevel + (B*4) - DefenderLevel) * 5 )%
B = item bonus (usually is 1). If you use freezing in ranged attack, then AL is decreasing by 6 and chance divides by 3. Duration = (chance - roll) * 2 + 25 frames, Durability ranges from 25 frames (1 second) to maximum of 250 frames (10 seconds).
In Diablo III, Freeze is still associated with Cold damage, and can only happen as result of certain Cold skills (mostly ones used by Wizards). The guaranteed shatter on death is gone and replaced with Critical Hit mechanics: a victim killed by a Cold Critical Hit is shattered into pieces of frozen flesh (or whatever substance they are made of).
Players can be frozen just like monsters, but the duration of all types of duration is reduced by the Crowd Control resistance or by the reduces duration of control impairing effects stat. Ice Climbers can even make target completely immune to Freeze and Immobilize effects. Being immune to Cold damage (that is, wearing the Talisman of Aranoch) does not prevent the character from being frozen. In addition, most skills with guaranteed Freeze have a limit on how often they can Freeze the same target (usually once every 5 seconds).
Frostburn gloves have a chance to Freeze target with any Cold damage for 1 second. Another item with an interesting synergy for Freeze is Rimeheart, which may inflict massive Cold damage on a Frozen enemy.