It is unclear how the custom of reading the Haftarah started. Some say (Levush, Shulchan Aruch 284) that it originates from 168 B.C. when Antiochus IV Epiphanes, King of Syria desecrated the Second Temple in Jerusalem and wanted to eradicate the Jewish faith. In order to do so, he decreed (amongst other decrees) that Jews would be prohibited to learn Torah, and that of course included the Torah reading in the Synagogue. It was then that the Haftarah came into being as a substitute to Keriat Hatorah, since reading from the books of Prophets was allowed.
Hence the name Haftarah from the root Patar (exemption), meaning that the Haftarah reading exempts from the Keriat Hatorah duty. When the decree was finally cancelled, the custom was continued.
Others (Shiboley Haleket 44) say that originally people used to stay in the synagogue after prayers to study the Torah, Prophets and Mishnah every day of the week. However, when times were financially hard, people devoted more time to making a living and therefore study was neglected. As far as the Torah was concerned, there was still the Keriat Hatorah in the synagogue, but the study of the Prophets was completely abandoned. In order to amend this situation, the reading of the Haftarah was introduced in addition to the Torah reading.
In the time of the Talmud (3rd-5th centuries) the Haftarot were not fixed. People just read from any place in the Prophets that was relevant to the weekly Torah portion. It was later, we don’t know exactly when, that a fixed portion was allocated to each Torah portion. Today there are still different customs regarding certain Haftarot (Sephardi and Ashkenazi).
The Haftarah has to be read from a book, as opposed to chanting it by heart. Although some congregations have a custom to read the Haftarah from a scroll, many read from a book (as opposed to a hand-written scroll on parchment, which is required for the Torah reading).
The Haftarah is read after the Torah Parasha has been completed. The person called up to the Haftarah is called the Maftir. In order to indicate that the words of the Prophets are not of the same importance as the Torah itself, after the Parasha is completed, the Maftir is called up to read again the last three verses (no less) of the Parasha with the Torah blessings. Hence the name Maftir for that section.
Before reading the Haftarah, the Maftir recites a blessing. After the completion he recites three other blessings and on Shabbat and festivals, four.