The Arm is a character that appears in "You're a Damn Good Cop, Jim Gordon". After Clayface gets his arm dismembered during a heist, his disembodied hand gains sentience and becomes a discrete entity until Clayface reabsorbs him.
The Arm appears just as Clayface's right hand does, a mound of clay in the form of a hand with five fingers. However, he also has a mouth with crooked teeth and yellow eyes of his own, and two stumpy legs at his base.
While Clayface and Harley Quinn were fleeing Wayne Enterprises, Clayface's hat was blown off. When he went to retrieve it, a security door slammed down on his wrist and severed his arm, which disintegrated into normal clay. Clayface wanted to retrieve his arm, but Harley insisted on departing, so The Arm was left behind. Separated from Clayface, it reformed into a separate being with all of Clayface's memories but significantly less maturity.
The Arm was then found by Batman, who bagged it for evidence and eventually brought it to the GCPD to give to Commissioner Gordon, who was going through a tough time now that he's discovered his wife has been cheating on him. Determined to prove his usefulness, Gordon nevertheless began to interrogate the sniffling Arm about Harley Quinn's whereabouts, although The Arm only wanted to "get back to my dad". Gordon then suggested talking about Clayface, but The Arm insisted that he's "not supposed to talk to strangers".
Changing tact, Gordon tried to establish some rapport by introduced himself as "Jim", a "hardworking detective with a beautiful wife who won't touch me even though I treat her like a queen and do everything for her. Everything!" He then realized he got sidetracked and tried to refocus on the subject at hand, but The Arm pointed out that "she doesn't sound like a friend to me", pleasing Gordon. He still tried to insist that it wasn't what they were there to talk about, yet The Arm suggested that "maybe it should be, Jim", so Gordon took The Arm to the roof to talk about it.
Jim, you sound like a really good cop.— The Arm
Gordon told The Arm all about his troubles, even about Batman taking away the Bat Signal, and The Arm comforted and supported him, calling him his friend. Gordon was overjoyed by this, but still tried to leverage it to finally convince The Arm to reveal Harley Quinn's location. The Arm asked him if he was simply pretending, and although Gordon admitted that he was initially, he said they were true friends now because The Arm didn't judge him. The Arm simply said "you can't judge what you don't understand", and then agreed to reveal the location, but was interrupted by the arrival of Harley Quinn and Clayface before he could do so.
They both demanded that Gordon turn over The Arm, who begged Gordon not to let them take him because he was "epidence". Gordon assured The Arm that he was far more than just "epidence", and that he wouldn't allow them to take him, at which point Gordon spun around and started shooting at Clayface and Harley. Gordon declared that they'd have to kill him to get The Arm, because he was "my friend. My only friend", and that they'd have to "pry this hand out of my cold dead… hand". The Arm agreed that they should just "leave us alone", so Harley came out from behind cover with the device she stole from WayneTech, threatening to "blow this whole fuckin' place sky-high" if he didn't relinquish The Arm.
Rather than give in, Gordon simply shot at Harley, who pressed the button in a panic and vanished into thin air, shocking everyone. With Harley gone however, Gordon was able to hold Clayface at gunpoint, who tried to reason with The Arm, saying that he only wants him to be "safe and happy". Batman himself then arrived with Harley Quinn, and although Gordon initially said that The Arm was his new friend, Batman apologized and Gordon immediately dropped The Arm to hug Batman and reconcile. He then urged The Arm to return to Clayface, who sniffled and whimpered in abandonment. He ran to Clayface, saying "I'll never forget you" until Clayface reabsorbed him and erased his identity.
It is revealed in the episode "Inner (Para) Demons" that Gordon had himself tattooed in remembrance of The Arm.
The Arm thinks and acts in much the same way that a young child might, with Clayface noting that he'll do anything for approval. He avoids talking to strangers, but is quite open to making friends, even with people who don't have his best interest at heart. Which shows he has extremely low self respect.