Mixed Lighting is a lighting situation where more than one type of light is present. It can be intentional, or a problem which needs to be solved either on location or when post-processing. An example of an intentional use of mixed lighting is the use of strobe and continuous lighting together in a studio setting for special effects. This combines a long enough exposure to show motion with the frozen image provided by flash. An example of undesirable mixed lighting is the use of a tungsten light in a room with daylight coming through a window. No matter what white balance you choose on your camera, because the two light sources are a different color your resulting image will have parts that are either blue / cool, or orange-red / warm.
What is Mixed Lighting in Photography
Photographically speaking, mixed lighting refers largely to the color of two different light sources, moreso than any other aspect of differing sources of light such as flash versus natural light, or bright versus dim, etc. Even though technically, any two different light sources could be considered "mixed".
Daytime ambient sunlight, for example, is roughly 5500 Kelvin white balance temperature. Indoor lighting that is roughly similar to the traditional Tungsten color temperature, on the other hand, will be in the vicinity of 3000 Kelvin white balance. (See also: Color Temperature.)
Mixed lighting conditions can also be created intentionally, such as using a colored gel on a flash for a special effect. Gels can commonly be orange-red or "warm", blue or "cool", as well as more "for effect" colors such as green or pink or any other color.
When shooting in mixed light, if you match your white balance to one of the light sources, that light source will appear neutral in color while the other light sources show their true (relative) color. For example, shooting an indoor + outdoor scene with daylight white balance set would render the outdoors "normal", while the indoors would be rendered with whatever warm color is coming from the lamps indoors. Oppositely, if you set your white balance to an indoor setting, (incandescent, tungsten) then that light would appear more neutral, while the outdoor light would appear extremely blue.
If the difference between two light sources is great enough, (dusk light is far more blue than normal daylight, for example) ...then splitting the difference between the two light colors would result in both very warm colors indoors, and very cool colors outdoors.
When processing a raw photo that is captured in mixed lighting, the same constraints apply when setting your overall white balance. If you match one light source, the other will exhibit its relative color. To fully correct this and match light colors, localized brushes and/or selective color adjustments would need to be used, to cool down or warm up any undesirable colors.
Oppositely, if a special effect is desired, pushing the white balance further in one direction can result in a very dramatic colorization.