The basic laws of kashrut are in the Torah and their details are explicated in the oral law, contained in the Mishnah and the Talmud. A common misunderstanding is that the Jewish dietary laws have to do with cleanliness; this is incorrect. The laws have to do with Levitical laws of ritual purity and holiness, sanctification.
The term "kosher" is frequently used in a metaphorical sense to mean "acceptable" or "approved". It is also part of some common product names. For example, "kosher salt", which is a particular form of salt in large crystals that makes it particularly suitable for preparing meat in accordance with Kashrut law, and is not meant to imply that the salt itself is kosher in the original sense (although it is, as is normal table salt, sea salt, and just about any other form of salt).
Meat
Meat, or foods that contain meat products, are not kosher unless the animal the meat comes from is kosher, and is prepared with specific methods. Kosher animals must have cloven hooves and chew their cud, so beef, for example, is kosher but pork is not. They must also not eat dead things (like vultures or catfish). Kosher fish must have both fins and scales, so shellfish are trayf as well. There are also strict laws forbidding the mixing of foods such as meat and dairy products.
Dairy
Milk, and milk-derived products, from kosher animals is itself kosher. In theory, then, all milk from cows is kosher. In practice, many Orthodox Jews use only "Cholov Yisroel" milk and dairy products; this label means that the milk that has been under constant rabbinical supervision from milking to bottling, to make sure that it is not adulterated with the milk of a non-kosher animal. In the past this was a serious issue; today this is not a practical concern in the USA or in most western countries; As such, most Modern and Centrist Orthodox rabbis, and all Conservative rabbis, have ruled that FDA supervision is sufficient for milk and dairy products to be considered automatically kosher.
No mixing of meat and dairy
One of the primary rules in kashrut is that milk products and meat products may not be eaten together in the same meal, much less cooked together. Jewish law thus mandates a set of fence laws that prevent this from ever happening: Jewish homes have two complete sets of silverware cookware, cups, and dishes. One is for milchik (milk) dishes, and one is for fleishing (meat) dishes. Foods that contain neither milk nor meat are called pareve (The Hebrew term), parve (the Yiddish term); both of these terms translate as "neutral".
The Reform Judaism and Reconstructionist Judaism movements do not mandate observance of the laws of Kashrut. Instead, they advise the study of Kashrut, and suggest that people follow those particular rules that the individual feels increases the sanctity of their life. As a result, there are some Reform and Reconstructionist Jews who do keep kosher.