The Haftarah (often written as Haphtarah, Haftorah or Haphtorah) is a section of the Prophets - the Nevi'im section of The Tanakh. This is the section that starts with Joshua and ends with the twelve minor prophets. It is read as part of the Kriat HaTorah service in the Synagogue on Shabbat (the Sabbath) and festivals, after the main Torah reading. Despite the name seemingly including the word Torah, it is actually from a different origin and means "conclusion".
The origins of reading a section of the Prophets go back to Roman times, when Jews were forbidden from reading the Torah. To somewhat compensate for this, a section of the Prophets was picked for each week and each festival, that taught a similar lesson, or had a similar theme, to the Torah reading for that week. And even now that we do read the Torah, the tradition of reading the Haftarah continues.
To avoid people thinking that the Haftarah reading is as important as the main Torah reading, the person being given an Aliyah (called up to the reading) for the Haftarah is actually called up for a reading from the Torah first. This is the Maftir (from the same P-T-R root as Haftarah - P and F are the same letter in Hebrew). On a normal Shabbat, the Maftir reading is the last 3-5 verses of the main Torah reading. On certain special Shabbats (eg when Rosh Hodesh falls on Shabbat), and on festivals, the Maftir reading is something related to the special day, and the Haftarah reading is also something appropriate for the day.
One of the main things a Jewish boy can do for his Bar Mitzvah is be called up to the reading of the Torah. And (at least in the UK), it is quite common for him to be called up for Maftir and then read both the Maftir and the Haftarah. This is actually a bit ironic, given that in fact you can be called up for Maftir and Haftarah even before you're Bar Mitzvah! However, the reason is this. The main Torah reading (including the Maftir) is very hard to learn, as the Torah itself is written with no vowels etc. The Haftarah, on the other hand, is usually read from a Chumash (or a book containing all the Haftarahs), but that can be printed with the vowels and note indications. So for those boys whose Hebrew isn't great, they only need learn a few verses from the Torah, and can then read the Haftarah from a normal book. Boys who are more capable at reading Hebrew will often read more (or all) of the main Torah reading, but are still often called up for Maftir so they can read the Haftarah as well.