Superman is the Trope Maker for the genre of modern superhero comic books. While there were super-powered characters before him, none of them included all of what are now seen as the "classic" Comic Book Tropes the way Superman did. Every superhero ever written since follows in his footsteps.
The Golden Age of Comic Books also had a secondary trend of vigilantes with ill-defined omnipotent powers, who'd fight street crime without much regard for ethics. See Dr. Fate, Green Lantern, the Spectre, Black Widow, and so on.
In the Silver Age, all comics followed trends, to the point where it was an in-joke among comic writers and fans, for example an EC story about hiring Jack Kamen includes the line, "Jack, you old son! I haven't seen you for two trends!" EC followed trends religiously for a while, then started their own, Horror Comics! This in turn led in part to The Comics Code (boo! hiss!).
Marvel Comics' success, particularly with Spider-Man, the first teen superhero, had many publishers trying for a more teenager-friendly product; sadly, these often faded into Totally Radical.
On the subject of Spider-Man, his comics were also one of the first to illustrate the "normal" side of the hero, along with Fantastic Four. Rather than filling the issue with one action sequence after another, part of the issue would illustrate Peter taking on everyday tasks such as getting to work on time, experiencing relationships, dealing with school bullies, and so on. Even nowadays, polls and streets interviews indicate that the main reason people like Spidey so much is because "he's a regular guy like the rest of us." It has since become standard for comics to portray the everyday side of the superhero, with the character, like Peter, being portrayed as someone the target audience can relate to. Unfortunately, under worse authors, this often results in myriad forms of This Loser Is You.
Before All-New, All-Different X-Men, most superheroes were WASPs. X-Men pioneered the concept of diversity (at least the token kind) in superhero comics.
And it's even Older Than They Think; the first international multiracial hero team was Cyborg 009, though it didn't achieve the success in the West that the X-Men did.
The X-Men, along with the New Teen Titans, set the gold standard for angst and melodrama in superhero comics, as well as telling more personal, character-driven stories. The widely-reviled Justice League Detroit was DC's attempt at ReTooling the JLA into an X-Men clone.
Nobody who's written Batman in the past 20 years has been able to escape the influence of Batman: The Dark Knight Returns. This case is particularly hilarious because the single greatest influence on Batman's character wasn't even canon. Even Batman's entry in the All-Star series, which was supposed to throw out all the complicated backstory and let the DC heroes have more Silver-Age-style adventures, was written by The Goddamn Frank Miller himself, and Batman was even more cranky and psychotic than ever.
In the early 80's, mainstream American comic books lagged behind some of their British counterparts which featured more sophisticated and literary dialogue and story concepts. Then, after Alan Moore reinvigorated DC's poorly selling Swamp Thing, DC editors quickly signed up other emerging British writers such as Jamie Delano (Hellblazer), Neil Gaiman (The Sandman) and Grant Morrison (Animal Man). This proved so successful that the "British invasion" of DC continues to this day.
This was parodied in the Doctor Who New Adventures novel Sky Pirates, which reveals Bernice Summerfield to be the author of a bizarre Vertigo-style comic called The 45 Second Piglet; said comic having been commissioned simply because she was in a big building in New York with a British accent.
Bernice was created by Paul Cornell, a British writer. Guess who he writes comics for now?
Isaac Baranoff's Horndog inspired a number of knock-offs. Baranoff even got in on the act himself by introducing Here, Wolf, which was not substantially different from Horndog, except for having human characters (it since differentiated itself though).
A large portion of The Dark Age of Comic Books was in some ways an extended attempt at following the leader by creating comics in the vein of The Dark Knight Returns and its contemporary, Watchmen, in an attempt to reflect the complexity and depth of these works. However, many critics — including, amongst others, Alan Moore, writer of Watchmen — accused them of only copying the superficial details, mainly represented by the Nineties Anti-Hero, rather than the storytelling complexity and experiments with medium that these works pioneered, with the result that most comics of this period were no more deep or original than the earlier works they were moving away from — they were just nastier.
Batman: Year One was the UR Example of origin stories in the more recent eras. Now both Marvel and DC produce Year One stories, with varying degrees of success, although none of them could match Batman's.
This article suggests that most of the nostalgic turn of recent superhero comic books can be tied to the popularity of Kurt Busiek's Marvels.