Is Godzilla a Monster Or A Lizard?

Technically, none. It has always been either a person in a very elaborate costume (as in all the Japanese Gojira films) or a computer-generated image (as in the American-made Godzilla film).

But if you're asking what lizard Godzilla is modeled after, the original Japanese Godzilla was a cross between a Tyrannosaurus Rex and a Stegosaurus (I believe the designers have gone on record saying this), and the US Godzilla looks to me to be basically a marine iguana with Jay Leno's chin added.