Short answers, pulled from the story.
Kashrut is the set of Jewish dietary laws that determine which foods a person may eat and how those foods must be prepared according to Jewish religious law. Food that may be eaten is called kosher, meaning "fit," and food that may not be eaten is called treif. Most of the basic laws derive from the Torah's books of Leviticus and Deuteronomy.
Kosher mammals must both chew their cud and have cloven hooves, which excludes the camel, hyrax, hare, and pig. Kosher fish must have both fins and scales, and all shellfish, reptiles, amphibians, rodents, and birds of prey are forbidden. The only kosher invertebrate is certain types of locust.
The rule against mixing meat and milk derives from a broad interpretation of the Torah's commandment not to cook a kid in its mother's milk. Meat and dairy may not be served at the same meal, cooked in the same utensils, or stored together, and observant Jews keep separate sets of dishes and wait between one and six hours after eating meat before consuming milk.
Shechita is the kosher method of slaughter performed by a certified slaughterer called a shochet, who severs the trachea, esophagus, carotid arteries, and jugular veins in a single quick cut using an ultra-sharp knife called a chalaf. The cut causes rapid and massive blood loss, since the Torah forbids consuming blood. The animal's body is then checked for 70 categories of disqualifying conditions called treifot.
A hechsher is a proprietary symbol or certificate showing that a product complies with kosher law after inspection by a rabbi or rabbinic organization. One of the most common in the United States is the O-U, a U inside a circle from the Union of Orthodox Congregations, while Britain uses logos such as KLBD from the London Beth Din and MK from the Manchester Beth Din.
Menachem Lubinsky, founder of the Kosherfest trade fair, estimates as many as 14 million kosher consumers and 40 billion dollars in sales of kosher products in the United States. The Orthodox Union claims that a kosher product placed next to a non-kosher competitor will do better by 20 percent, and buyers include Muslims, Seventh-day Adventists, vegetarians, vegans, and the lactose-intolerant.
In colloquial English, kosher often means legitimate, acceptable, permissible, genuine, or authentic, and the Babylonian Talmud once used it to mean virtuous, calling Darius I a "kosher king." The term also appears in uses like kosher cell phones, which are stripped-down devices with limited features sold by some Orthodox retailers.