Answer:
   
   Note: Much of this information is summarized or extracted from Steve
   Weintraub's Kashrut Class on Meat and Milk,
   [5]http://www.chelm.org/jewish/kashrut/l3.milk_and_meat.html, with
   permission.
   
   The Torah commands us three times (Exodus 23:19, Exodus 34:26 and
   Deuteronomy 14:21) not to cook a kid in its mother's milk. The Talmud
   widens this to the complete separation of milk and meat, including
   bird meat. Why do we do this?
     * Rambam attributes it as a prevention of an idolatrous and
       superstitious practice.
     * Some attribute as a discouragement from a cruel practice.
       
   There are many possible reasons, but we should not, in general, try to
   find reason in Torah prohibitions. It is not for us to fathom G-d's
   reasons in telling us to do something; it simply should suffice that
   G-d asked us to do it. And, by doing it, we are reminded of G-d's
   commandments and the fact that we are Jewish.
   
   Everything in the Torah is considered to have meaning; thus, the
   rabbis have determined that the triple repetition of the warning in
   the Torah means three different types of prohibition:
    1. You may not cook such a mixture
    2. You may not eat such a mixture
    3. You may not benefit (in any way) from such a mixture
       
   This was interpreted very strictly. Meat products were not permitted
   to come into contact with milk products in any way. Food, and the
   utensils used to cook and serve food, were divided into three
   categories:
     * Milchig (or chalav): Food containing milk, or utensils used with
       such food.
     * Fleishig (or basar): Food containing meat, or utensils used with
       such food.
     * Pareve (or stam): Food that is not derived from milk or meat and
       is not cooked with a milchig or fleishig utensil. This food can be
       eaten with either milk or meat (although in certain circumstances
       use of a milk or meat utensil will render the food milchig or
       fleishig). Pareve foods include all vegetables, grains, fruits,
       eggs and fish. Originally birds were considered pareve (when was
       the last time you saw a chicken give milk?), but the Rabbis ruled
       that bird meat should be considered fleishig to avoid confusion.
       
   Milchig and fleishig food can not be eaten together. There is a
   waiting period (depending on your tradition, as previously discussed
   in the FAQ--see the answer to question [6]6.6) between eating meat and
   milk. No waiting period is required after eating milchig food before
   eating fleishig food, but one should rinse one's mouth. There is a
   rule that one must wait an hour after hard cheese for just this reason
   (a hard cheese being defined as a cheese that has sat for six months
   or more).
   
   Along with not eating, the two types of food can not come in contact
   while cooking, nor can utensils used for such cooking come into
   contact. This has typically led to the "two sets" one sees in Kosher
   kitchens: Utensils and plates for meat, and utensils and plates for
   milk. Add in Passover, and you'll see that a Kosher household has four
   sets of dishes, at minimum. These are all stored separately, and
   typically are marked so as to clearly differentiate them. Food cooked
   in the wrong pot is not kosher.
   
   Of course, there are some exceptions:
   
    1. Glassware. Glass was considered non-absorbent by the Rabbis. As a
       result glass can be used interchangeably between the two types of
       food, as long as it is cleaned well. The custom among Askenazic
       Jews is to soak the glass 72 hours before interchanging, the
       Sephardic say soaking is unnecessary.
    2. Sinks. First, stainless steel sinks are preferred, as they can be
       rekashered (porcelin sinks are porous, and are difficult to make
       kosher). If there is a double sink, one half can be used for milk
       and one half for meat. If this is impractical, then you treat the
       sink as treif (non-kosher). Utensils and food should then only
       touch it if they are going into the dishwasher. Individual dish
       racks (meat, milk) should be used in the sinks to avoid contact.
       In treif sinks, you may not soak utensils or food; a separate
       kosher basin must be used.
    3. Ovens and ranges. It is not necessary to have separate ovens and
       ranges for meat and milk. If the same oven is used for both, great
       care should be taken to avoid spills and splatters. Both types of
       food should not be cooked in the same oven at the same time.
       Grills used for one can not be used for the other without
       kashering. When cooking on top of a range food should be covered,
       and great care needs to be taken to avoid splatters. It is best to
       specify the meat and milk burners, covering the unused side with
       foil. Many people avoid this problem by having separate ovens.
    4. Dishwashers. A dishwasher can be used for both meat and milk
       dishes, but not at the same time. Dishes should be well rinsed
       before being put in the dishwasher. Between the two types of
       loads, a rinse cycle should be used. It is also preferable to have
       separate racks for meat and milk. Many people address this problem
       by using the dishwasher for either milk or meat, and hand washing
       the other.
    5. Towels Towels that are freshly clean can be used either with meat
       or milk. Once they are used for one or the other, they must be
       washed before use with the other. It is best to have different
       towels for each to avoid confusion.
       
   For traditional Jews, the prohibition from benefiting from a mixture
   is interpreted strictly, so buying a cheeseburger for a non-Jewish
   friend is forbidden. Note that the mixing of milk and meat only
   applies to meat made from kosher animals (so you can buy your friend a
   ham and cheese sandwich), and the stricture is stronger for cooked
   food than uncooked food. Milk and meat that accidentally mixed, but
   not cooked, can be sold or given away. Milk and meat that is mixed and
   cooked must be thrown out.
   
   You'll find the separation of meat and milk to be followed in the
   traditional movements, as well as the Conservative movements. In
   Reform Judaism, these rules are only followed in those households that
   find the observance of Kashrut a meaningful practice.