The Muppet character who is a blue, elephant-like monster is Thog.
Thog is a huge, blue, full-bodied monster with a horn, drooping ears, a broad cow-like face, and a hippopotamus-like snout. His design combines features from several animals, which is why “elephant-like” is a useful description without being a literal species classification. Thog is also known for having a gentle, sweet disposition.
He first appeared in The Great Santa Claus Switch, a 1970 television special featuring Muppets created by Jim Henson. Thog later became part of the wider Muppet character roster and appeared in The Muppet Show and subsequent Muppet projects.
The character’s enormous size is part of his visual appeal. He was originally presented as about 9½ feet tall, although a later appearance in The Muppets depicted him as roughly 11½ feet tall. A common mistake is choosing Sweetums, another large monster; Sweetums is shaggy and orange-brown, while Thog is distinctly blue and has a horn.