With exceptions, the Wolf has never officially been named. Red has called him Wolfie in her songs. In The Tom and Jerry Comedy Show, He is known as Slick Wolf and in Tom and Jerry Kids, he is known as Slick McWolf.
He first made his appearance in Droopy's first cartoon, Dumb-Hounded in 1943. In the story, the Wolf, who is a killer, escapes from prison and keeps running into Droopy no matter where he goes, scaring him. In the end he is captured and Droopy gets a lot of money for his capture.
The Wolf is most famous however, in the short, Red Hot Riding Hood, which is the first in the trilogy mentioned above. In the short, Little Red Riding Hood, Grandma, and the Big Bad Wolf get fed up with how the story was never changed and the narrator, annoyed complies. He changes the setting so that Grandma lives in a penthouse, Little Red Riding Hood is a gorgeous dancer/singer at a nightclub, and the Wolf is a womanizer heading towards the nightclub. He soon sees Red dancing and singing and he falls madly in love with her and goes really nuts. He asks her out but she refuses and goes to visit her Grandma. The wolf heads to Grandma first and to his horror, Grandma instantly falls in love with him and chases him all throughout her home. The Wolf ends up falling out the window and says he's through with women and would kill himself before he looked at another girl. Red shows up and he kills himself but his ghost falls in love with Red and goes nuts.