Initially it was adherence to the Torah . There are several written laws in the Torah about what is OK to eat. Leviticus 11.3,9,13-19,21,29-30 and Deuteronomy 14.6,9,11-18 detail what much of it. There are some rules about slaughter as well. The one which is the best known is the commandment that you shall not "scathe a kid in its mother's milk". There is an anecdote in the Talmud about a Rabbi who no one cared for much who used to eat chicken with cheese. Since chickens don't produce milk, it's not difficult to avoid scathing it in its mother's milk. As a result of that one guy, it was decided that it's within the letter of the law, but not the spirit. Much of it is argued to have been law for health reasons, though some of it is clearly more in reference to animal rights. Part of that is acknowledging that the creature you're about to eat sacrificed itself so that you can continue living. It is believed in Jewish tradition that the Torah is no longer in heaven, so it is ours to modify as we see fit. The Talmud is essentially just that. A bunch of rabbis getting together and deciding what's right and wrong. One particular discussion (about the oven of Akhnai) of kashrut (the dietary laws) makes this point in direct defiance of what was clearly God's will. http://www.mechon-mamre.org/jewfaq/kashrut.htm is a fairly good resource on the teachings in the Torah.

As another answer states, observant Jews follow kosher laws "because God said so" in the Old Testament. I won't repeat what has already been said well, but I will mention that I heard a different account of why chicken and milk cannot be mixed even though chickens don't give milk. As a chicken parmesan-loving, non-Jewish girl trying to respect the practices of the kosher-keeping Jew I love, I have often been frustrated by this rule and demanded explanations! My boyfriend, who has a Talmudic reference to explain EVERYTHING, told me that two rabbis once debated this issue. One rabbi said that chicken/dairy combinations were okay because chickens don't give milk, and thus you can't risk boiling a chicken in its mothers milk. The other rabbi argued that, while this may be true, well-intentioned Jews could confuse chicken with other types of meat. They could accidentally break the milk-meat law by mistakenly thinking they were eating chicken, when they might actually be eating the flesh of a mammal that should not be combined with dairy. And the concept of "better safe than sorry" won out, and chicken was added to the list of meats that can't be eaten with milk products. However, as my boyfriend was quick to reassure me, tradition has established some sort of compromise where chicken/dairy is forbidden now but will be allowed in the afterlife! I'm no Talmudic scholar and I can't provide a reference, but I share this story because (a) I like it and (b) I think this "better safe than sorry" philosophy explains a lot of now-standard practices that might not have originally been in the Torah, as well as the debates that Orthodox Jewish scholars are famous for having over seemingly minute details. An example of the former is the fact that Orthodox Jews who keep kosher will keep entirely separate and labeled sets of dishes/silverware for meat-based meals and dairy-based meals. This eliminates the risk that the two will ever come into contact, even accidentally.