Blend modes (Also known as Mixing modes) are used to define the way two or more images combine. There are many blend modes, some common ones include Add, Subtract, Multiply, and Screen.
A blend mode is a method for combining two colors. You can use a blend mode to blend a single pair of colors or two entire images.
Fundamentally, a blend mode is a mathematical function that takes two color arguments and outputs a new, blended color argument. Examples of blend modes include:
The Add blend mode, which simply adds the color channels. This has the effect of making colors appear lighter, as the resulting color is closer to white than the original colors (or equal, if one color is black). The Add blend mode can easily cause colors to clip past white. Mathematically:

Special values for the Add blend mode: black has no effect (adding zero); white produces white (adding one).
The inverse of the Add blend mode is the (rarely-used) Subtract blend mode.
The Multiply blend mode, which simply multiplies the color channels. This has the effect of making colors appear darker, because color channels are measured on the interval
, and multiplying any two numbers on that interval will always produce a number closer to zero than the original (or equal, if one number is 1). Mathematically:
Special values for the Multiply blend mode: white has no effect (multiplying by one); black produces black (multiplying by zero).
The inverse of the Multiply blend mode is the Screen blend mode.
The Screen blend mode, which inverts the images and then multiplies them, before finally inverting them again. The result is a lighter image, for the same reason that Multiply is darker, but then the image is inverted, so it will be lighter. Mathematically:

where S is defined by
These three blend modes are the most common blend modes, but there are many more as well. For more information, see the Wikipedia page on Blend modes.