When an item is Indestructible, it has no durability, therefore it has no repair cost and cannot be destroyed.
In Diablo I, it is a very rare mod on high-tier weapons and armor. The item's durability (not charges!) is frozen at highest possible value, and cannot be lowered. Therefore, an item cannot be broken and lost.
Jewelry is always indestructible by default.
In Diablo II, any item that comes with the Indestructible mod as a default, such as Uniques, cannot be ethereal unless it is also the default mod on it. Such as the Hell mode unique Ghostflame or the Assassin unique Shadow Killer. Some set items come with the Indestructible mod as well, but set items are the only equips (aside from Rings and Amulets) in the entire game that are never ethereal.
The Zod rune also grants any item it is inserted to the Indestructible mod, which is used in various situations for added defense and damage in certain items, such as an ethereal Hellslayer or the famous Breath of the Dying Rune Word.
In Diablo III, all items have durability (even jewelry), but any item can roll a secondary affix 'ignores durability loss', which is, in fact, same as Indestructible from previous games.
In addition, Spaulders of Zakara are widely known for making all of the bearer's items indestructible, effectively bringing repair costs to zero.
Since durability is lost very slowly in combat, an item cannot break permanently, and Hardcore characters usually do not care about loss of durability upon death, value of this property greatly diminished in D3, except for very early in game, when saving gold is important.