It's a fungus that looks like white powdery patches on leaves, stems, or flowers that later turns grayish brown. These are the usual causes:
The powdery patches are fungal spores and they multiply rapidly. You'll usually notice the mildew first on the top surface of older leaves. By this time, the disease has been inside the plant a week or longer. The affected leaves may turn yellow or brown and eventually fall off.
African violets and begonias are often susceptible because they prefer the high humidity that is ideal for fungus growth.
The easiest way to control the spread of this fungus is to cut off all the affected leaves and give the plant more light.
Move the plant away from other plants to improve air circulation and to prevent the spread of fungal spores.
You've probably heard of using baking soda as a treatment. This works well for prevention, but does not cure an affected plant. If you want to try this solution as a preventative measure, spray the plant with a mixture of 1 tablespoon of baking soda dissolved in 1 gallon of water.
Spray the affected areas with fungicide made for house plants. The 3-in-1 Insecticide/Fungicide/Miticide (shown here) works to control powdery mildew and other types of fungus and insects. Read the label carefully and follow the manufacturer's directions for use.