Orthodox Jews are religious Jews who follow halacha, traditional Jewish law and practice. Hasidic Jews therefore fall under this category as well, as halacha-abiding Jews. I must say, I wasn't familiar with the term "halachic Jew" until you used it, but I'd assume it means exactly the same thing as Orthodox Jew.
(Before the Jewish Reformation in the 1800s, the term "Orthodox Jew" did not exist - you were either an assimilated or converted Jew or a Jew who adheres to halacha.)
In the 1700s, Rabbi Israel b. Eliezer (d. 1760), known as the "Baal Shem Tov" (Heb. "Master of the Good Name"), began teaching from his homestead in Medzhybizh, Ukraine. The followers of the Baal Shem Tov became known as "Hasidim", which means "pious ones" in Hebrew. Hasidic philosophy is not something I can summarize here; it's such a large topic upon which volumes are written. Basically, Hasidim live like normal Orthodox Jews, except that they are extremely particular in their observance of Jewish law, and have incorporated mystical teachings into their daily practice. Rabbi Isaac Luria (d. 1572 in Safed, Israel), known as the Arizal, was a mystic. The Hasidim took his teachings and tried their best to reconcile them with the already familiar practices of Ashkenazic Jewry. Therefore, Hasidic prayer books began to emerge, which were different from the traditional Ashkenazic liturgy, as they brought in kabbalistic additions from the teachings of the Arizal.
When the Baal Shem Tov passed away, his prime disciple, Rabbi Dov-Ber b. Abraham (d. 1772), known as the Mezritcher Maggid (Yid. "Preacher from Velyki Mezhyrichi") sent his disciples to various places in Europe to spread Hasidism. Different "dynasties" of Hasidim arose from the towns the disciples were sent to. This is the origin of many of today's Hasidic dynasties.
Chabad was started in Lithuania by Rabbi Schneur Zalman of Liadi. A long time ago, there were many dynasties within Chabad, but after the Holocaust only one remained - Chabad-Lubavitch, the branch of Chabad Hasidim from the city of Lubavitch (Yid. "Lyubavichi"). This is the Chabad which exists today.
Other Hasidic dynasties, such as Gur, Satmar, Belz, Vizhnitz, etc. are named after the cities in which their teachers came from. Gur = Góra Kalwaria, Satmar = Satu Mare, Belz = Belz, Vizhnitz = Vyzhnytsia, etc.
Chabad specializes in outreach to unaffiliated Jews and religious Jews in remote areas. This behavior was encouraged by the last Lubavitcher Rebbe (a Rebbe is the leader of a Hasidic dynasty), Rabbi Menachem Mendel Schneerson (d. 1994). One obvious reason why Chabad is different from other Hasidic dynasties is its lack of distinct Hasidic garb shared by other dynasties, however, the philosophy of Chabad differs from that of other dynasties. But this is true throughout all dynasties (ie, Gur is different from Satmar). Entire books have been written about Chabad and its philosophy. I'd suggest if you want to learn more, you do some further reading. I've only provided the basics here. If you have any further questions, feel free to ask in the comments.
I didn't know what you knew and didn't know, so I summarized everything here. (To fellow Mi Yodeya users: what do you think of my summary? Comment below.)