Kiddush is a Hebrew word from the root k-d-sh which means to sanctify. Pronounced as it looks, or often "Kiddish". It is the same root as Kaddish.
It is most commonly used to refer to the bracha (blessing) said before drinking the wine at meals on Jewish festivals, including Shabbat (the Sabbath), which, despite occurring every week, is more important than any other festival except Yom Kippur.
The Kiddush prayer in the evening is the more important, and usually consists of two or three sections.
On Shabbat, the first section read is from Genesis 1:31 to 2:3. This deals with the end of G-D's creation of the world, and the fact that he then rested on the seventh day - the essence of the Shabbat. This is therefore appropriate for the Shabbat Kiddush. There is no equivalent reading on other festivals.
Following on from this is the normal Bracha for wine - Blessed are You, oh Lord our G-D, King of the Universe, Who has created the fruit of the vine.
There then follows the key part of the Friday night / Yom Tov evening Kiddush, where the day itself is sanctified. The blessing for Friday night thanks G-D for the Sabbath as a memory of creation, and as a memory of the exodus from Egypt. The blessing for Yom Tov deals with G-D having sanctified the Jews, and the holiness of the day.
After this blessing, everybody drinks from the wine before going out to wash their hands, making the blessing over bread and having dinner. Traditionally, "Kiddush Wine" is used. This is a sweet wine, which is most often identified with "Kosher Wine", even though there are some excellent dry Kosher Wines available. But most people still use sweet wine to symbolise that Shabbat is a sweet, relaxing time.
The Kiddush for Shabbat morning (lunch) is less important, but is euphemistically called "Kidushah Rabbah" - the Great Kiddush. Technically, all that is required is to say the blessing for wine, but it is often preceeded by appropriate biblical passages such as Exodus 31:16-17 which deals with the Children of Israel observing the Shabbat, or Exodus 30:8-11 which is the 4th Commandment - to keep the Sabbath. On festivals, a few verses are recited related to the festival.
Kiddush is also used to refer to what's best described as a small reception after a Shabbat morning service in Synagogue. Some Synagogues have one every week, some only when there's a specific celebration (eg a Bar Mitzvah). It is basically a small social event, but is called a Kiddush because it starts with the recitation of the Kiddush for Shabbat morning, as described above, usually by the Rabbi. Wine is provided, often whisky and biscuits, and sometimes pastries, hot drinks, fruit etc.