I believe Superman started out with a cape to basically keep the blanket he had as a baby with him in his costume.
Yes. Superman saves the world with his blankie.
As for other people in the DC verse, like Batman, his is far more practical. He uses it for gliding, to distract his enemies by sweeping it in their faces, to help him blend with shadows by wrapping it around him, covering the brighter parts of his costume. In some cases, he fireproofs it so he can use it as a shield against large amounts of heat, like when fighting Firefly. Continuing off the distraction aspect, it makes him appear bigger while fighting, with his cape sweeping around, his body vanishes within it, so it's harder to tell where he is to hit.
... I can't think of another hero who really uses a cape in mainstream comics.
In Marvel, you'd be hardpressed to find a person who still uses a cape. It's impractical in most cases. The only one I can think of is the Sentry. And he wears that partially to remind people that he's from an older time, so the cape harkens back to the old days when Superman pretty much defined the good guy stereotype.
... Still blanking on any other hero in DC and Marvel who uses a cape-THOR! Thor has a cape.
Though that's more of a fashion statement of Asgard. A display of formal regality. As a prince, it's a symbol of his stature. Hence why Odin also wears a cape: He's a king.
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