This article is from the of the muppets FAQ, by Craig Silverstein with numerous contributions by others.
The current best answers are "whatever" and "human," respectively.
Christopher Finch, in the almost-definitive "Of Muppets and Men," relates the following ad-libbed exchange on the backstage of "The Muppet Show": "Kermit/Jim Henson turns to Scooter/Richard Hunt and asks, 'What are you anyway?' 'My mother was a parrot,' Scooter replies. 'We don't know about my father. It was during the war.' 'Really?' says Kermit. 'Which side was he on?'" Finch describes both Scooter and Gonzo as "creatures of indeterminate provenance."
Continuing the trend, on the official Henson web site (at http://www.henson.com/ask/ask_archives_body.html#Goelz3) David Goelz talks coy about Gonzo, saying not even Gonzo himself knows. In many movies he's called a Whatever -- starting in "The Great Muppet Caper", where that's the marking on Gonzo's airplane crate -- and that's the answer we Faqmeisters usually give. (Though _Muppets From Space_, in which Gonzo is an unnamed alien species, muddies the picture still further.)
Note that the answer has changed over time: the Gonzo Muppet's first appearance was as a Frackle in "The Great Santa Claus Switch" (though clearly this is not the Gonzo we know and love today). Another alternative identification is in "Muppet Babies," in which Gonzo is sometimes called a "weirdo." Also, in "The Muppet Movie," Kermit says Gonzo looks "sort of like a turkey... but not much." (Judging by the Muppet turkey in "The Muppet Family Christmas," he doesn't look much like a turkey at all.)
As for Scooter, his uncle, J. P. Grosse, who owns the theatre, is clearly human. Despite Hunt's joking around, Scooter is probably a human too. While he's called a "gofer," that's his position and not his species ("gofer coffee," etc).