Do Jews have words to differentiate themselves from each other based on degree of Orthodoxy? I probably need to know them if they do.
The main divisions in the US are Reform, Conservative, and Orthodox. Elsewhere they may go by different labels. (E.g. I know Conservative is Masorti in Israel.) Reform is geneally considered the least observant, Orthodox the most, and Conservative somewhere in between. There is also a group that calls itself Reconstructionist. I've never met any of these SFAIAA, but my understanding is that they are mostly Jews by birth who celebrate the traditions but are non-Biblical.
If I was brought up as anything it was Reform, and when I returned to Judaism as an adult I started at a Reform Congregation. But I learned too much there and moved on to a Conservative Congregation which is really "Conservadox." (I.e. Our services are virtually identical to the Orthodox services, but women participate fully and microphones are permitted.) Most other Conservative synagogues that I have visited omit parts of the service, mostly to shorten it it appears to me. Some also add some words that remember Sarah, Rebecca, Rachel, and Leah, in addition to Abraham, Isaac, and Jacob. Some are led entirely by a rabbi and/or cantor. (At my synagogue, the rabbi never leads. The services are mostly lay led.)
As for divisions among the Orthodox, I am not an expert, but the major division seems to be between the Hareidi (Black Hats) and the Modern Orthodox. But there are lots of other divisions. Despite claims by each that they are most "authentic," no one stones adulteresses to death as is commanded in the Torah.
There is also a division in each group between Sephardic and Ashkenasi. The Sephardi have their roots mostly in the Muslim lands, and the Ashkenasi have their roots mostly in Eastern Europe.
ML/NJ
If you mean within the realm of Orthodoxy, the two main Orthodox groups in the US are the Orthodox Union and Agudath Israel. There are also independent hassidic groups.
OU covers a spectrum from ‘Modern Orthodox’ centered around Yeshiva University, to ‘black hat frum’ from European style smaller yeshivas.
Agudath is only ‘black hat’ traditional and includes some hassidic groups. Ner Yisrael Yehiva near Baltimore is a common place for Agudath education.
The late Rabbi Moshe Feinstein was a towering figure in American Jewry, reverenced by both groups.
I’m leaving out Sefardim, who in America are mostly more like Reform, but who nominally seem to maintain Orthodoxy in personal status questions.
Reform Jews are often very light on the Judaism, high on the social justice. Conservative Jews are more into actual Judaism and keep kosher etc at least in the synagogue. They are often still liberal in politics. Orthodox Jews try hard to follow the whole Torah as best they can, and are often not leftists. Reform Jews call themselves reform Jews and are proud of it. Conservative Jews won’t call themselves “conservatives” unless they are politically conservative, which is fairly rare. Orthodox often call themselves shomer shabbos Jews or something rather than “orthodox.”