Colloquially [sometimes pejorative, sometimes affectionate] kipa sruga. Adherents usually approve many aspects of secular culture, especially secular education, in addition to traditional Torah study. They tend to be Zionists. The precise term depends on the speaker – Rabbi Norman Lamm uses “centrist,” Rabbi Shlomo Riskin uses “cosmopolitan,” and Rabbi Emanuel Rackman uses “modern.” The Union of Orthodox Jewish Congregations, Yeshiva University, and the Rabbinical Council of America in some sense represent this group. In Israel, the Mizrachi organization is a well-known representative.
Colloquially, [sometimes pejorative, sometimes affectionate] “black hat” or “black”) suggests an Orthodox outlook in which the focus of life is Torah study, as is done in Lithuanian style yeshivot. Secular culture is either tolerated or criticized for its corrupting influences. This group tends to be “non-Zionist” in the sense that they love the land of Israel and its holiness (many spend years in Israel for Torah study) but are unenthusiastic about secular Zionism and Israeli secular culture. Agudath Israel tends to represent this group.
In Israel, this distinction is more a matter of attitude toward Zionism than of political affiliation or religious views. The Dati tend to be more supportive of Zionism, while the Haredi typically do not support the modern Jewish state. Please note that these are general positions; individual members may hold different views.