Maftir (Hebrew: מפטיר, "concluder") properly refers to the last person called to the Torah on Shabbat and holiday mornings: this person also reads the haftarah portion from a related section of the Nevi'im (prophetic books).
Informally the portion of the Torah read by or to the maftir is called the "maftir portion", or the "maftir" for short: in a Chumash the word "maftir" is printed in the margin at the beginning of that portion. (Accordingly, in those communities where the bar mitzvah acts as maftir, his readings are informally referred to as "maf and haf".)
On a normal Shabbat morning seven people are formally called up to the Torah, and a part of the week's Torah portion is read by or to each of them. The maftir is not counted among the seven, and is not formally called up by name: on the conclusion of the seventh reading the reader simply calls "maftir" (usually after reciting Chatzi kaddish) and repeats the last few verses to the maftir.
On Jewish holidays and certain special Shabbatot there are readings from two or more Torah scrolls. On these occasions, the maftir is called up by name, followed by the word "maftir", and the reading from the last scroll is read to him. On Tisha b'Av morning and fast day afternoons, the maftir portion is the third (and final) section of the portion.
After the Torah reading, the maftir says the blessings for the haftarah and reads it.
The maftir portion for Shabbat during Chanukah comes from Numbers 7, describing the dedication offering of the Mishkan (Tabernacle during the wilderness journeys) corresponding to the day of Chanukah where Shabbat occurs.
Shabbat Chanukah and two of the special Shabbatot (Shekalim and HaHodesh) sometimes coincide with Rosh Hodesh. When this happens, the portion for Shabbat Rosh Hodesh is read from a second scroll, then the special maftir portion for that special Shabbat from a third. Only the person called to the third scroll reads the haftarah, though the haftarah itself may contain verses appropriate both to Rosh Hodesh and to Chanukah or the special Shabbat.