Complements are colors arranged opposite to each other on the color wheel, but split-complements are colors on either side of a particular complement. The complement of yellow is violet, but the split-complement of yellow is red-violet and blue-violet.
A color scheme is a planned arrangement of colors for a work of art, home, brochure or other project that is purposeful. A split-complementary color scheme, therefore, is a planned design using three colors--the main color and two colors on either side of its complement.
Split-complementary color schemes include using red as the main color and accenting it with yellow-green and blue-green, or using blue as the main color and red-orange and yellow-orange as accents.
The purpose for any color scheme, including split-complementary, is to make a statement, whether you intend to create an overall mood or to emphasize certain areas within the design.
Consider using a split-complementary color scheme when creating a water scene, like Frederick Carl Frieseke's "Through the Vines," painted in 1908. The figure is rendered in red-orange, but she is surrounded by blue water and green foliage.