Chanting the haftarah in its minor key trop (cantillation) is an initiation rite passed through by virtually every barmitzvah boy. I remember investing so much effort in honing the public performance of the reading that I was entirely oblivious to its meaning until about a decade after my barmitzvah.
The haftarah is a selected passage from the Prophets that is read after the weeks Torah parashah. The haftarah is always thematically connected to the parashah except during this period, between Tisha bAv and Rosh Hashanah, when the haftarot are all about comfort and consolation.
The origin of the haftarah-reading custom is unclear. The most widely held view is that it originated during the rule of Antiochus IV (c.174-164 BCE), when Jews were forbidden toread the Torah publicly, so they read from the prophets instead. When the decree was lifted the custom continued.
On the other hand, Rabbi Samson Raphael Hirsch (1808-88) thought that the practice originated in opposition to sects which viewed only the Pentateuch as constituting the Hebrew Bible.
Haftarah is often pronounced haftorah, suggesting a mistaken belief that the etymology is related to Torah. Rather it is from the verb patar, meaning to leave, depart or conclude.