Foreground alpha is a rendering mode that sets the pixel alpha (opacity) based on if the pixel is considered to be in the scene foreground or background.
If you enable foreground alpha, then when you save your image to PNG or EXR, or any format that supports an alpha channel, then transparency information will be included in the saved image. This allows for easy compositing of the objects in your scene over a background image.
For an example, see the Compositing with shadow pass tutorial.
Here is a render made with foreground alpha enabled. The grey checkerboard pattern indicates alpha zero (fully transparent) pixels.
The foreground alpha option is found in the render settings section: