Is the word "fight" a noun? And what is a verb? I think it is but I'm not sure. I've looked at a couple websites with lists of nouns, but it's not on there.

It can be either a noun or a verb depending on the context.

It is correct to write 'a fight', which means the word becomes a noun. It can be used for example in a sentence "I saw a fight between two wolves in a documentary film."

But it can also be written in a sentence like: "Wolves fight one another to set a hierarchy in the group.". In this sentence 'fight' is used as a verb.

And It is also a gerund like: "fighting is not a good habit". Here it is a gerund, As it is also a verb and used here as a subject and noun.