When or where do Legendary dragons appear in the world?

Legendary variants of dragons were added for high-level players; they begin to appear at around level 78. Once you get that high, they can be found just like any regular dragon (although they are not guaranteed to spawn): randomly in the wilderness, at word walls, et cetera.

Note that Legendary dragons were added by the Dawnguard DLC, so you'll need that.

Once they spawn, fighting them is basically the same as any other dragon (except they are stronger and have a wider variety of shouts). There's more information on them at the UESP wiki for Skyrim.

There is a guaranteed spawn at Arcwind Point.

From Elder Scrolls Wikia:

A legendary dragon can also always be found at Arcwind Point perching above a sarcophagus that contains either a Draugr Deathlord or a Dragon Priest.

This spawning point still requires the The Elder Scrolls V: Dawnguard DLC and level 78+. If the player dose not meet these requirements, another type of dragon will spawn at Arcwind Point.

You generally have to be lv 78 and have dawnguard installed, but their are people who have encountered them anywhere in-between lvs 50 through 77. I've seen the videos but it is extremely rare and they can spawn anywhere they can spawn in whiterun, windhelm, winter hold, excetera.