What Makes Food Kosher?
At its most basic level, keeping kosher means following three rules:
1. Dairy products (i.e. foods made with milk) and meat are not eaten together. For purposes of kashrut, "meat" includes mammals and poultry but not fish.
2. Certain animals (like pigs and rabbits) and fish (like catfish, eels, or shellfish) are never kosher.
3. In order for the meat of a kosher animal to be kosher, the animal must be killed and the meat prepared in a particular way.
The first two rules come directly from the Torah, more or less. The idea that poultry counts as meat was a later addition, as were the particulars on how to shecht (kill and prepare) meat. Over the last two thousand years, additional rules have been developed to reinforce the three central rules, such as having separate dishes, utensils, pots & pans for dairy and meat.
There is a common misconception that a rabbi must bless food in order for it to be kosher -- not so. However, shochets (kosher butchers) and mashgichim (kitchen supervisors; singular is masghiach) require training to ensure the rules of kashrut are followed faithfully, so often rabbis take on these roles. Packaged foods that have been prepared in a kosher manner and supervised by a masghiach carry a heksher, a symbol indicating that the food is kosher. Each kashrut certification agency has its own unique heksher.
Today, there are almost as many variations on kashrut as their are Jews, from glatt kosher (which enforces stricter rules about the quality of a cow's lungs) and cholov yisroel (which requires stricter monitoring of milk products to prevent the mixing of different animals' milks) to eco-kashrut (which is concerned with our food's impact on the environment) and magen tzedek (literally "shield of justice," indicating that food was prepared with fair labor practices).
If you're interested in learning more about kashrut, check out MyJewishLearning.com's extensive section on Jewish Dietary Laws.