"...like the pulps before them, comic books and comic strips contained all the elements of the superhero — the powers, the mission, the identity — but it took Siegel and Shuster to put them all together into Superman." —Peter Coogan
Superheroes owe so much to Superman that they use his name—if Superman had the name of his first imitator, Wonder Man, we would speak today of wonderheroes.
Coogan's attempt to list the defining elements is probably as good as any, so I'll expand on it:
1. The powers are an extraordinary ability. Even human superheroes like Batman are far, far better at what they do than almost anyone else.
2. The mission is a commitment to justice. The "hero" in "superhero" matters. Monsters like the Heap and the Hulk are sometimes called superheroes, but they usually do what they do out of necessity, not choice. Superheroes may wish someone else could do their jobs, but they follow the code that is Stan Lee's best line (though the idea is at least as old as the Bible): "With great power comes great responsibility."
3. The identity is a public persona that includes a distinctive name and appearance, which is separate from a private persona that lets the person pass in public without being easily recognized. Even the superheroes who don't have secret identities are best known in their superheroing roles—like other celebrities, they won't necessarily be recognized in common clothes.
The Oxford English Dictionary says the word was borrowed from the French and has examples of it being used as a noun before Superman's appearance in 1938:
1899 Daily Mail 29 Sept. 4/4 M. Clémenceau suddenly burst out with, ‘All the world knows that Colonel Picquart is a hero, but..if Colonel Picquart is a hero, Mathieu Dreyfus is a .’
1917 ‘Contact’ Airman's Outings 211 The of the war.
1924 N.Y. Times 16 Dec. 28/1 It is all very well to have a in such narratives, but this man among men should occasionally show signs of being human.
1937 Thrilling Wonder Stories Aug. 120/2 The strip started off very well, but I must agree with others that it is rapidly degenerating into the juvenile antics of a musclebound .
And as a compound:
1916 Harper's Weekly 11 Mar. 245/2 A bill... This was a bill to give more than their regular pensions to soldiers who had been more heroic than their duty called for.
Merriam-Webster's definition points out that the superness of a superhero only calls for superheroes to be extraordinarily good at what they do:
a fictional hero having extraordinary or superhuman powers;
If we use the three categories that Coogan and others cite to say Superman is the first superhero, we can eliminate many of the popular candidates. The Scarlet Pimpernel and Robin Hood did not have costumes—most predecessors of the superhero wanted to be able to blend in rather than be recognized. Spring Heeled Jack was a villain. John Carter and Tarzan used the names they had grown up with and did not have private lives that were separate from their public ones. Mandrake was a magician—if you say he's the first superhero, you have to explain why Merlin or Circe is not.
But masked heroes like the Scarecrow (who appeared in 1915, several years before Zorro) meet the three requirements. The Lone Ranger and the Phantom both abandoned their private lives, but they still kept their identities a secret and sometimes mixed in society in street clothes.
It's fine to be inconsistent. If you're only talking about comic books, Superman is the first superhero. If you include newspaper comics, the Phantom is. If you include popular fiction, either the Scarecrow or Zorro is—the Scarecrow appeared first, but in his original appearances, the Scarecrow is a less virtuous figure than Zorro. But if you're just speaking colloquially, the first superheroes are the oldest heroes of myth and legend. It's all good.
Related: Skintight costumes—why the Domino Lady may have been as important as the Phantom in the creation of Superman
ETA: While I was writing this, comments were being made on a Facebook post, Wrong on several counts, that are relevant, and since I wrote this, it's become part of the discussion there.