The Penguin is Oswald Cobblepot, Gotham City criminal, information broker, nightclub operator, occasional umbrella artillery enthusiast, and a man who absolutely notices when somebody fails to take him seriously.
Unlike many Batman enemies, Penguin does not usually need madness, superpowers, or elaborate philosophy to justify his place in Gotham. His greatest weapon is understanding how the city works. Cobblepot builds influence through money, secrets, criminal connections, respectable fronts, and a willingness to make himself useful to dangerous people until he has enough leverage to become dangerous himself.
The theatrical details remain part of the fun. Depending on the story, Oswald may favor tuxedos, monocles, trained birds, trick umbrellas, or an entire aesthetic built around the fact that Gotham once gave him a cruel nickname and he decided to trademark the insult through sheer spite.
Adaptations push him in different directions. Batman Returns presents a grotesque outsider emerging from Gotham’s sewers, while other versions emphasize the sophisticated nightclub owner, political operator, or organized-crime boss. Those interpretations can differ dramatically without needing separate Character identities.
The durable Penguin is the same creature underneath: underestimated, resentful, calculating, and extremely good at turning Gotham’s existing corruption into a business opportunity.