This is a VERY big question. I’m not aware of a good online source to explain it all. I will do my best, focusing on American Orthodox Jews. (I’m one.) It will be long.
The key thing to remember is that, especially when not talking about Hasidic groups, these are not discrete categories. All of them shade one into the next, resulting in the creation of more and more category labels over time to try to neatly categorize individuals, especially for the purpose of setting up single people for marriage. IMO, all such efforts at labeling are somewhat useful but inevitably doomed. To give you an idea of the complexities of this, try this breakdown of the broad categories at one Orthodox dating site: click here, then on ‘How are the Orthodox categories defined?’, then on the link they give. I am reasonably ‘Modern Orthodox Machmir’ in terms of practice, but don’t really see a category there that fits my worldview.
That said: Divide the American Orthodox community into two broad groups: Modern Orthodox and Ultra-Orthodox. Both groups seek to maintain a lifestyle that adheres to traditional interpretations of Jewish law (halacha). However, they differ in their approach towards the outside world. MO tries to balance outside influences with living a life in accordance with halacha, while the UO generally tries to remain more cloistered, preferring to live with as little contact with the rest of the world as possible. This manifests itself in different approaches towards secular education (which MO values highly, with essentially everybody in an MO community has a college degree and frequently postgraduate education, while in the UO world they get what is deemed enough to comply with regulations and earn a living, although this varies significantly), the media, the internet, and many other things. Additionally, the two groups view the role of women very differently; all Orthodox groups restrict the role of women in public ritual, but the UO world is very focused on gender segregation, modest attire for women, and idealizing women as mothers and homemakers, while the MO world is generally more integrated and encouraging of women’s career. The MO world still places restrictions on women’s dress/behavior, generally discourages dating except for the purposes of marriage, often forbids physical contact between unrelated people of the opposite gender, and frequently encourages single-sex education, especially for high school. However, gender separation in the UO world is much more wide-ranging, including separate education from nursery onward, separate seating at all functions such as lectures, weddings, etc., and a general discouraging of any unnecessary social contact between unrelated people of the opposite gender. While all Orthodox groups are very family-oriented, and there is no social concept of opting to remain single, or childless by choice, a typical MO family might have 3-5 children, while an UO family will generally have 7-12; UO groups are generally discouraging of birth control.
Of course, there are many people who may identify as Orthodox but who are not committed to being adherent to halacha. It’s a social identifier, rather than a truly religious one. IMO, this group is significantly smaller, in terms of both percentage and numbers, than it was 30-50 years ago. There’s been a huge increase in Jewish education in the interim, and those who identify as Orthodox tend to be much more knowledgeable than they used to be in the US.
There is never any proselytizing of non-Jews. In fact, halacha requires that one try to discourage conversion candidates.
There isn’t a formal proselytizing effort between the two broad groups. However, as working in Jewish education doesn’t pay well, the more highly educated MO group tends not to work in Jewish schools, even MO ones. As such, the teachers, especially for Judaic subjects, are frequently UO, and they may push their viewpoint in ways both subtle and not. A year in Israel learning in yeshiva/seminary is universally encouraged, and it’s not unusual for an MO young person to ‘flip out,’ becoming much more observant and/or much more UO in his/her orientation in a short period of time. Also, Chabad proselytizes to all non-Hasidic Jews, to some degree, but see below - they’re a special case.
Within the UO world, you can broadly divide into two groups, Hasidic and non-Hasidic. This is a more discrete division, although there are those who are ‘Chasidish-ish,’ who like the culture of Hasidism but who may not closely follow the dictates of any one Hasidic sect. The non-Hasidic are referred to as ‘Yeshivish’ or ‘black hat,’ for the black fedoras that have become near-mandatory in those circles for males over bar mitzva. The Hasidic can, themselves, be broken down into two groups: Chabad, aka Lubavitch, and all other Hasidic groups.
Hasidic groups largely emigrated to the US after the Holocaust. Each sect follows a single grand Rebbe, admiring him to such a degree that his table leftovers are considered blessed. The grand Rebbe position is generally hereditary, although it may pass to a son-in-law (who may have been selected for this) rather than to a son. I can think of two current Hasidic groups that have opted not to replace a beloved Rebbe after he died, and who are thus without a single leader: Breslov, who have been rebbe-less for 200 years, and Chabad (see below.) Aside from Chabad, they are very inward-focused, with the entire sect generally living in either one place, or a short list of communities (e.g. Belzers live in Jerusalem, Borough Park, Brooklyn, and I think Montreal.) The lingua franca is generally Yidish; boys, whose Jewish education is prioritized, may receive very little English education and may speak English poorly and with a heavy accent, even if their family has lived in the US for 4-5 generations. The sexes are even more rigidly separated in non-Chabad Hasidic groups than in other UO groups; for example, wedding celebrations are generally held in two separate rooms, rather than in one large room with a divider. The men tend to have distinctive dress, in the style of Eastern European gentry of the eighteenth and nineteenth centuries, and I find that when people think of “Orthodox Jew,” they tend to be picturing “Hasidic man.” Some groups get along quite well (e.g. they may marry across sect lines), and some really don’t. I’m not plugged in at all to the intricate world of Hasidic politics, and can’t really give you a good overview, but it can get fairly ugly, especially in Israel. This tends to be political/financial rather than purely religious, although they may have slightly different interpretations of halacha (and yes, the narcissism of small differences definitely applies here.)
Chabad is a Hasidic group, one currently riven by a division between those who revere the most recent Rebbe, but acknowledge his death in 1994, and those who still consider him the Messiah. (Frankly, non-Chabad Orthodox Jews find the latter group quite disturbing.) The most recent Rebbe charged his followers with bringing all Jews closer to observance of Jewish law and Judaism. These are the Hasidic Jews who may have asked you if you were Jewish at some point. They do not proselytize outside of Judaism, but can be quite aggressive within it. To that goal, they’ve founded centers around the world, and may live far from any established Orthodox community at all. They obviously choose to interact with many outside of their own groups, are more MO in terms of gender segregation, and are big fans of the internet and technology, which allows them greater outreach. Other Hasidic groups tend to be uncomfortable with them, and they don’t have much social mixing between the two.
I have never heard of inter-Hasidic-group proselytizing.
There are also non-Chabad inter-Jewish prozelytization efforts (aka ‘kiruv’), from both UO and MO organizations. My husband became religious in his teens through NCSY, an MO group.
Yeshivish UO Jews are also very inward-focused, although the degree can vary quite significantly by community. The lingua franca is generally English. The major communal value is Torah study, to the point that men are encouraged to spend many years focusing exclusively on full-time Talmud study in yeshivas, while they are supported by their parents and wives. They do not follow a dynastic Rebbe, but over the last 30-40 years, there has become greater communal pressure to follow the dictates of rabbinic leadership of the yeshivas, rather than of the communities.
If I may be less detached than I’ve been trying to be above, in general, MO respect the Judaism of UO, and many even idealize it as more authentic/traditional (which I would dispute), but UO uses the word ‘Modern’ as an epithet. While you will often see UO rabbis invited to speak at MO synagogues/teach in MO schools, you will almost never see the reverse.
That’s the best quick (ha!) overview I can give you. Does this answer your questions?
at theology; in fact, except for sporadic efflorescences in the Medieval Scholastic period, examining the ontology of God has always been viewed as getting way too close to the fire.