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.
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.
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).
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.