Chemicals are great for cleaning but the wrong combination can result is dangerous gases or violent reactions. The big problem is a lot of these chemicals are found in most households. On their own a lot of these chemicals are safe and have few safety risks. It is only when mixed together do big problems arise.
Bleach and Ammonia
One of the worst combinations of chemicals (and most dangerous) is bleach and ammonia. The resulting concoction gives off the toxic vapor of choramine. Immediately vacate the area and seek professional help if you accidentally mix these two together.
A more common problem is mixing bleach and vinegar. Both are very common in homes and vinegar is often mixed with baking soda for cleaning purposes. But bleach and vinegar will release chlorine gas, which is toxic. While the resulting mixture can make a better disinfectant, it is never recommended. The danger far out weighs the benefits.
Base and Acid
Whenever a base (e.g. ammonia) and an acid (e.g. vinegar) come in contact a vigorous reaction will occur that releases heat. The magnitude of the reaction will depend on the chemicals. Some of the worst reactions will start splashing around and can be a threat for your skin or eyes.
One reason people accidentally do this is to try to clean tough spots. The thought is mixing more chemicals might help clean better. The problem is a base and acid will react and result in a neutral chemical that is worse for cleaning. To top it off the reaction can give off harmful fumes.