Hebrew term that literally means "The Reading (Kriat) of the Torah". Used to refer to the public reading of the Torah in Synagogue.
The Torah itself is the Old Testament, handwritten by a Sofer on parchment. It is then rolled up on two wooden poles, "dressed" in a fancy cover (often velvet) and decorated with breastplate, a "yad" (a pointer) and one or two crowns on top, often all made of silver.
The Torah is read on the following occasions.
Shabbat morning
The Torah is read every Shabbat morning. It is split into 54 portions, and most weeks one is read. Due to the way the Jewish calendar works, there are between 51 and 55 Shabbats in the year. However, if a Yom Tov falls on Shabbat (as usually happens), then the reading is appropriate to the Yom Tov. To make up for the lack of weeks, there are certain portions that can be doubled up.
The weekly reading is split up into seven sections, and at least seven people are given an Aliyah to the reading - they are "called up" by their Hebrew name, to make a blessing before and after their section is read. Sometimes, it is split up further to allow more people to be called up. Traditionally, the person who was called up would read the portion. However, to avoid embarassing people who don't know how to, nowdays most Synagogues appoint one person each week to learn and read the whole weekly portion.
Shabbat afternoon
The first section, split up into 3, of the next week's portion is then read on the afternoon of the Shabbat. This is in preparation for the next week.
Monday and Thursday mornings
The same reading from the previous Shabbat afternoon is read again as part of the weekday morning service on Monday and Thursday. This dates back to when these would be the market days, and so Jews never go more than 3 days without hearing a public reading of the Torah.
Festivals
On festivals, an appropriate reading to do with the festival, split up into 5 portions is read. This takes precedence over the Shabbat reading - a festival falling on Shabbat has the festival reading, although it is split up into 7 sections like Shabbat.
For example, the readings for Pesach are to do with the ten plagues and the crossing of the Red Sea by the Children of Israel.
After the main reading, a further short reading, the Maftir is read from a second Sefer Torah, detailing the sacrafices in the Temple that were made on that festival.
On the intermediate days of Pesach and Succot, a smaller reading for 4 people is read.
Fast Days
On fast days, the reading is concerning the Golden Calf. This was seen as the first proper repentance by a people as a whole, and fast days are seen as an auspicious time to ask G-D for forgiveness.
The reading is split up into three, and is read in both the morning and afternoon services.
Rosh Chodesh
The Rosh Chodesh (New Month) reading is about the sacrafices on the start of a new month. Four people are called up.
The whole procedure
If you are a visitor to a Synagogue, the wardens (who run the service) may ask you for your Hebrew name, to be called up. If you aren't Jewish, tell them! If you don't want the honour, also tell them. However, while they will often try to give visitors an honour, there are certain people who have a right to be called up (eg a bridegroom, a father who has just had a baby etc), so this isn't always possible.
If you are actually called up, you recite the following blessings. These are usually written on a small plaque on the Bimah, and often transliterated as well.
You say: Barchu Et Adonai, Ha'Mvorach - Blessed is G-D, the blessed one
Everybody else says: Barach Adonai, Ha'Mvorach, L'Olam Va'ed - Blessed is G-D, the blessed one, for ever and ever
You repeat that line.
You then say: Baruch Atah Adonai, Eloheinu Melech HaOlam, Asher Bachar Banu MiKol Ha'Amim, V'Natan Lanu Et Torato, Baruch Atah Adonai, Notein Hatorah - Blessed are you, oh Lord our G-D, King of the Universe, who has separated us from other people, and given us your Torah, Blessed are you oh Lord, who has given us the Torah.
The reading will be read. If you can, follow along.
On completion, say: Baruch Atah Adonai, Eloheinu Melech HaOlam, Asher Natan Lanu Torat Emet, V'Hayei Olam Natah Betocheinu, Baruch Atah Adonai, Notein Hatorah - Blessed are you, oh Lord our G-D, King of the Universe, who has given us your True Torah, Blessed are you oh Lord, who has given us the Torah.