The laws of kosher require that in addition to not eating them together, we wait a specified period of time between eating meat and eating dairy.
After eating dairy and before eating meat, eat something pareve, which does not stick to the palate. Then rinse your mouth, or take a drink, and wash your hands. In addition, many have the custom of waiting a certain period of time -- a half-hour or an hour. After eating certain hard cheeses, a six-hour waiting period is required (see Why the extended wait between eating "aged" cheese and meat?).
After eating meat foods, we wait six full hours before eating any dairy. The six-hour waiting period is standard for all Jews, except those groups which have halachically established other customs.
If a small piece of meat is discovered between the teeth, remove it and rinse the mouth, but an additional waiting period is not required (even if six hours have elapsed since eating meat). If even the smallest amount of food is chewed or swallowed, the full waiting period becomes necessary.
If food is tasted but immediately eliminated from the mouth before chewing or swallowing, then no waiting period is required. One should rinse the mouth well.
Meat and dairy foods may not be eaten at the same meal, even if they are in separate dishes and even if the waiting time elapses.