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The laws of Kashruth define what is and is not Kosher
The word Kosher comes from the hebrew root כשר which means fitting or appropriate. Food which is fitting or appropriate for a Jew to eat as defined by the laws of Kashruth(from the same root) is therefore termed "Kosher". The laws of Kashruth are the oral laws that the Rabbis set down based on the written laws specified in the Torah regarding that food which a Jew is permitted to eat1.

A few general notes (in no particular order)

  1. While most animals must be killed (schechted) in accordance with Jewish law, (with perfectly sharp knives, covering the blood of wild animals etc.) Fish are considered prepared and ready to eat simply by being caught. Which is why kosher sushi can be prepared.
  2. The gid haNasheh, the sciatic nerve, must be removed, Jews are forebidden to eat it directly in the text of the Torah itself. (bereishith/genesis 32:33 Yaakov/Jacob was wounded in the sciatic nerve in his fight with the angel.)
  3. The subtleties of the laws of Kashruth differ greatly among different religious Jewish traditions. (Sephardi and Ashkenaz for example)
  4. There are some very specific insects which we are permitted to eat. There is much halachic discussion as to which insects these are as well as when we are permitted to eat them.
  5. Lastly, in the absence of all Kosher food, a Jew is permitted to eat non-Kosher food for survival purposes, but only in extremis. In other words if a Jew ended up on a desert island with no edible source of food except wild pigs, s/he would be permitted to eat them. Still they would be required to kill it first due to the restrictions of not eating of a living animal. (eyver min ha Hai) This is certainly not a loophole for people to live their entire lives eating non-kosher because they chose to live in a place with no easy access to kosher food. According to Jewish Law they would be required to move as soon as it is at all feasible.
  6. Also, there are many times throughout the year where Jews are required to eat or drink specific foods or types of foods.

An entirely separate time-bound category of kashruth

  1. During the holiday of Pesach/Passover which is seven days long, (eight outside of Israel) all leavened bread is prohibitted. Leavened bread is defined as any flour of any or all of the five grains (wheat, barley, spelt, oats, rye) mixed with water that has had enough time to rise (usually 18 minutes) before it has been fully baked.
  2. While all vegetables and fruit are permitted, one may not eat from the fruit of a tree in the first three years of its fruit production.
  3. It is forebidden to eat any fruit or vegetable that was planted or harvested in the land of Israel during any seventh year of the Shmita cycle.
  4. There are certain fast days upon which all food is forbidden. (People who are sick or pregnant are an exception.) They are, Yom Kippur (10th of Tishrei), Tisha B'av (9th of Av), Asarah B'Tevet (10th of Teveth) and Shiva Asar B'Tamuz (17th of Tamuz). There are other fast days but these are the big four.
  5. There is an accepted Jewish custom not to eat meat for the three weeks leading up to the fast day of Tisha B'av.

More complex special-case subtleties of Kosher Law

  1. In the time of the Temple, it is forbidden to eat of any produce prior to the taking of the holy tithe.
  2. According to the Rambam/Maimonides in the midst of a war a soldier is not liable for eating non-kosher food found in an enemy camp; even if they have a ready supply of kosher food. This is a very extreme case (from what I have heard it is) meant to illustrate that during warfare one cannot expect soldiers to exercise total control over themselves.
  3. A Jew cannot eat food cooked (by or for a Jew) on shabbath/Saturday, while they can eat food on a holiday prepared specifically for that holiday. (In addition to food prepared during the week of course.)

while this writeup is meant to cover (in a general way) most of that which was left out in early writeups, it is in no way exhaustive. There is far far far more to be said on the halacha, the Jewish law, of preparing and eating food. Please inform me if I have left out anything drastic or if I have misconstrued or simply erred regarding Halacha. As always, when in doubt consult your Rabbi.
1. This intro was added at Gritchka's suggestion. It is redundant for clarification purposes. As someone who is unfond of redundancy, I apologize.