Maftir, is a Hebrew word meaning Conclusion. It is from the P-T-R root, as in Hebrew, F and P are the same letter. It refers to the concluding part of the reading of the Torah in the Kriat HaTorah service on the Shabbat (Sabbath) and festivals. See also Haftarah.
In most cases, the Maftir is simply the last 3-5 verses of the main reading repeated. However, there are a number of occasions where a different Maftir is read. In these cases, it is usually to take a 2nd Sefer Torah out of the Aron HaKodesh, as otherwise it can take some time to roll the one you've been using to the correct place.
The special Maftirs are as follows.
Festivals
All the major festivals - Rosh Hashanah, Yom Kippur, Succot, Pesach and Shavuot have a Maftir reading from the Sedra (portion of the Torah) of Pinchas. This contains passages detailing the extra sacrafices that were made in the Temple on each festival - hence it is appropriate.
Rosh Hodesh
Rosh Hodesh, the New month, has a special Maftir also read from Pinchas, and again detailing the extra sacrafices that were made on Rosh Hodesh. Interestingly, there were also extra sacrafices made every Shabbat in the temple (with respect to the regular daily sacrafices) but this section isn't read every Shabbat - although it is included in the special Maftir for Rosh Hodesh.
Chanukah
The Torah is read every day of Chanukah, with a different reading for each day. On Shabbat Chanukah, the reading is read as a Maftir. It is taken from the Sidra of Naso and details the dedication of the first Temple, when the princes of the 12 Tribes of Israel each bought an offering. Chanukah is only 8 days long, however, so on the 8th day, the last 5 days (from 8 to 12) are all read.
The special Shabbatot before Pesach
There are four special Shabbatot in the run-up to Pesach.
- Shabbat Shekalim - the reading is from Exodus and is about G-D asking Moses to take a census of the Jewish people by each one giving a half-shekel. The money was then used to build the sanctuary.
- Shabbat Zachor - this reading is from Deuteronomy and is G-D commanding us to remember what Amalek did when he attacked the Children of Israel in the desert. Interestingly, this is the only reading from the Torah that is mandated by itself, all other Torah readings are Rabbinical. Hence it is considered particularly important to hear. This is always the Shabbat before Purim, appropriate because Haman is seen as a descendent of Amalek.
- Shabbat Parah - This is from Numbers and deals with the ritual known as the "Parah Adumah", literally the "Red Heiffer". This is a mysterious process in which a red-haired cow was sacraficed to purify those who had become spiritually unclean after contact with a dead body.
- Shabbat HaChodesh - This is from Exodus and is the Shabbat nearest to Rosh Hodesh Nissan. It deals with the first direct commandment from G-D to the Children of Israel - to start counting months.