The Joker's Face is the result of the Joker having had his face surgically severed from his head. Since then it had shifted many hands from character the character in a variety of different stories.
When the Joker became obsessing over destroying the Bat-Family, he decided to arrange for his face to be surgically removed. According to the Joker, he did this to symbolize how even without his, "Mask" he was still the joker where-as Batman's Robins and Bat-Girls' identities were only mask-deep.
To see this through, Joker arranged for the serial-killer known as the Dollmaker to remove his face surgically in Arkham Asylum. Dollmaker sent his, "Son" to do the procedure only for Joker to brutally murder him by tearing out his throat with his teeth. Dollmaker later went in person to as Joker refused any other way and after Dollmaker finished, the two murderers arranged to celebrate his rebirth before leaving the clown's face on the cell-wall for Batman to find.
Death of the Family
About a year later, the Joker broke into the GCPD building where he cut the power and murdered several police-officers with his bare hands by snapping their necks in the dark from behind. He would psychologically torture James Gordon before stealing his old face back and vanishing.
Joker would use straps, clips and pins to retrofit the flesh of his old face into a mask which he wore over his now exposed tendons, hiding how he now looked. Joker would remain in this mask for the remainder of his plan to destroy the bat-family, a mission which ended with a farce dinner within the very caves of the Batcave.
During a fight with Batman in the caves, Batman began turning the tables on Joker by calling Joker's own identity and allure into question. When Batman bluffed about having known Joker's real name, Joker attacked Batman with a taser joy-buzzer to the face before letting him say it and as a result fell down a subterranean water-fall. As he fell, his mask became dislodged and floated down after him to a presumed watery grave.
The Joker's Daughter
The Joker's face would find its way through the subterranean rivers of Gotham to its sewers where it was found by the deranged Duela Dent. Duela took the face and began to wear it as a mask amongst the sewer nomads of Gotham City under the new name of, "Joker's Daughter"
Duela would go on to track down the Dollmaker and arrange for him to surgically graft the severed face of the Joker onto her own where it has since remained.
Since the Joker's apparent death in the bat cave, his body happened upon a pool of the substance dionesium which healed wounds and revived the dead. Within the pit, Joker grew a new face which he would go on to use during his endgame.
Appearances in other Media
In this series, the show's Joker analogue was a boy named Jerome Valeska. When Jerome seemingly died, his corpse went through a procedure which attempted to revive him. When the procedure seemed to fail, a cult decided to instead remove Jerome's face due to his criminal reputation.
However it came to be revealed that Jerome had indeed been revived as he was forced to wear bandages around his head before learning that the leader of the cult which stole his face was using it as a mask. Jerome would kill the cult-leader to get his face back and later have it grafted back on.