Even a small trace of a non-kosher substance — as little as 1/60th (1.66 percent) of the food's volume, and in certain cases, even less than that — will render an otherwise kosher food not kosher. By the same token, utensils that come in contact with hot food will absorb its "taste" and subsequently impart it to other food.
For example, a loaf of bread baked in a pan smeared with shortening that contains a small percentage of lard, fruit juice pasteurized in the same machinery as non-kosher milk, or a vegetarian dish prepared in a restaurant kitchen with the same utensils in which a non-kosher dish was cooked earlier — these would all be regarded as non-kosher if the proportion is greater than the permissible percentage.
It is for this reason that separate utensils are used for meat and milk, and that a reliable kosher certification is needed for foods processed or prepared outside the home.
I don't see an article in this kosher section addressing how to handle situations when mistakes happen. A person must not only know how to make their kitchen kosher, but also what to do once they have a kosher kitchen and a mix-up occurs. It is important that people know to immediately consult a rabbi, and how to not make things worse.