What is the Haftarah?
The Haftarah is the passage from Nevi'im, the books of the Prophets, read in the synagogue after the Torah reading on Shabbat, festivals and fast days. The word comes from a root meaning “to conclude”: the Haftarah closes the reading.
How the Haftarah connects to the parashah
Every parashah has its Haftarah, chosen by the sages for a link to the theme of the Torah portion: a parallel episode, a key word, a promise that echoes. In Parashat Vayera, for example, Avraham's hospitality and the announcement of a son are echoed by Elisha and the woman of Shunem (Melakhim II 4).
The Haftarah is read by the maftir, who is called to the Torah to repeat the final verses of the parashah. He recites blessings before and after, including one thanking G-d for choosing “good prophets” and their words, spoken in truth.
Special Haftarot of the year
At certain times the Haftarah follows the calendar rather than the parashah. In the three weeks before Tisha B'Av the three Haftarot of rebuke are read, and in the seven weeks after it the seven Haftarot of consolation, opening with Yeshayahu's “Nachamu, nachamu ami”. On the Shabbat between Rosh Hashanah and Yom Kippur, Hoshea's “Shuvah Yisrael” is read.
There are also special Shabbatot such as Shekalim, Zachor, Parah, HaChodesh and Rosh Chodesh, each with its own Haftarah. In many cases Ashkenazi and Sephardi custom read different passages or passages of different length.
Why read the Prophets
A traditional explanation, cited by Abudarham, holds that the Haftarah began when a decree forbade the public reading of the Torah: the sages read passages from the Prophets tied to the parashah's theme instead, and the custom remained after the decree fell.
Whatever its origin, the Haftarah shows the Torah alive in the history of Israel: what the parashah teaches, the prophets bring back before the people in another generation.