Let's knock these babies out:
(1) Gothic Fiction. Dark and stormy night? Check. Horrid monster? Check. Attempt to excite sublime feelings like terror and awe? Check and check. Sounds pretty gothic to us.
(2) Tragedy. Everyone dies at the end. Oh, you want to get more technical? Okay: Frankenstein shows us a great (or, well, pretty good) guy brought low by his own pride, or hubris. Definitely a tragedy.
(3) Science Fiction. This is where it gets interesting: Frankenstein is often considered the first work of science fiction. What's key is that the science isn't just window-dressing: the whole point of the novel is to explore heavy questions about What It All Means, where "It" can be loosely translated as "science, fate, free will, nature, and humanity." You know, easy stuff like that.