The phrase pretty much means what it sounds like it means.
Bad language lesson! Japanese verbs go at the end of the sentence. Desu is simply the verb “be.” Wa indicates the subject of the sentence, which in this case is ore, a first-person pronoun. So: “I am an American ninja.” With the word “American” in English, and the word “ninja” in… er, whatever.
Ore, meanwhile, isn’t just any old first-person pronoun. Japanese has many, many first-person pronouns, depending on the speaker’s age, gender, status in relation to the audience, and so forth. One dictionary defines ore as a “boastful first-person pronoun.” Chris's cooperating teacher Chiho explains that she “doesn’t like” this word, although her students do. Jason explains to me that you’re only allowed to use ore when you’re both (1) drunk and (2) hitting on the listener. And, really, what else would an American ninja be doing?