"There's something different about you...don't tell me...new haircut?"- The Penguin, Batgirl #53
Penguin: "Pity. I was hoping she might be as hot as that little blonde number that was traipsing around as Robin ever so briefly. What happened to her? She get killed too? Oh why do the girls always meet such tragic-WAH!
(Batgirl headbutts him)
-Batgirl #59 "Fresh Blood Part 4: Settling Up"
Real name: Oswald Chesterfield Cobblepot
Stephanie met the Penguin as Robin in Batgirl #53. The Penguin was coordinating an illegal arms shipment and using children as his couriers, including one little girl Stephanie and Batgirl had to save so she didn't get killed by Oswald's "customer".
Cassandra and Stephanie tracked down the Penguin and Stephanie was so furious that he had put children's lives in danger (an anger exacerbated when Cobblepot bragged that when she put him in jail he'd use his lawyers to get back immediately) that she gave him a sound beating and had to be stopped by Cassandra grabbing her hand and saying, "Enough."
The Penguin later mentioned Steph in Batgirl #59, part of a crossover between the Batgirl and Robin books, after her death, where he mocked Tim about her, only to be interrupted by a headbutt from Cassandra in the above exchange.
Later, in an attempt to take control of the Gotham Underworld in the absence of several major members of Batman's rogues gallery, Penguin had employed a female agent to do his dirty work in taking out Tobias Whale. He was very impressed with this agent and, for reasons unknown, told her to dress up in a Spoiler costume he had...for some reason. (Gotham Underground #5) She succeeded in all of her duties, including infiltrating Oracle's Metropolis headquarters and blowing it up. (She also beat the snot out of Tim and Babs, with the aid of what appeared to be a cloaking device. Tim was "saved" by Wildcat, who knocked Spoiler out, but she escaped before Tim could unmask her.) (Gotham Underground #3)
Spoiler abandoned Penguin at a crucial point in his plans (Gotham Undeground #7), meaning she was likely Steph undercover.
Meanwhile, over in Robin, Penguin was attempting to eliminate Violet and Robin. When Stephanie interfered, Penguin ordered his henchmen to kill her too, though he was unsure of her superhero name and was only told she was a "butt kickin' chippie in purple". He didn't succeed in eliminating any of them, but it doesn't seem to bother him.
Shortly after that, Tim cut a deal with the Penguin in exchange for the criminal finding Batman, who was missing and possibly insane. Stephanie expressed her disapproval of this.
Later, she stole the money Tim used to bribe the Penguin from the criminal, leaving a Robin-a-rang. She inexplicably decided to use this money to hire Scarab to help her reclaim Gotham.
Find out more about The Penguin here: