So I’m watching a Batman cartoon, and I’m wondering. Does the Penguin have any superpowers? If not, then what’s the deal?
Many, if not most of Batman’s enemies have no super powers. If you think that The Penguin is bad, take a moment to think about The Riddeler, this guys special quirk is that he tells Batman what he is going to do. At least The Peguin has a bunch of cool umbrellas.
In the recent cartoons, though, (as well as Danny Devito’s portrayal in the second film) the Penguin is portrayed as some kind of mutant, or at least with unusual (though not superhuman) characteristics/birth defects of some kind. The original comic-book Penguin was called such because of his interest in bird-based crimes and his penchant for wearing tuxedoes (i.e. “penguin suits”) all the time. There was never any indication that he had special powers, beyond a rather impressive ability to train birds. Also, the comic book version was never portrayed as being psychologically twisted, as Devito’s version seemed to be.
If anything, Devito’s sewer-dwelling outcast is the exact opposite of the dapper little social climber with good taste originally written in DC comics.
Short answer: no.
Long answer: noooooooooooooooooooooooooooo.
Ooooh, scary! 
Batman also doesn’t have superpowers. That’s what makes Batman unique amongst superheroes.
He has the power to suck.
Penguin, as has been noted, has no superpowers.
He does have an impressive arsenal of tricked-out umbrellas (umbrellas that are flame-thrower, umbrellas that are machine guns, umbrellas that spray knock-out gas), and has trained birds to commit crimes.
As has been noted, this is goofy as all hell (but fun, which is what counts). So, in recent years, he’s been reinvented as more of a Godfather type. He mostly stays in his nightclub and runs Gotham’s undeworld through intermediaries. His power is being really well-connected.
Most of Batman’s rogues gallery has no powers, just gimmicks and mental defects. The Riddler must prove his intellectual superiority by leaving clues to his crimes. Two-Face is driven to commit crimes related to duality or the number two, his power is super-indecisiveness (he uses a two-faced coin to determine whether he’ll do a crime or not). The Joker’s power is that he’s Batship insane.
According to his official comic book origin story, the Penguin is a little fat guy who’s obsessed with umbrellas (his mother always made him carry one when he left the house) and with looking good (hence the tuxedo).
He has been reshaped by recent retcons into a sort of crime overlord kind of fellow, rather than someone who routinely goes toe to to with the Batguy.
Ah, the umbrellas. I guess that qualifies as a superpower in a Batman cartoon/comic. Thanks, all.