Typically, you wouldn't define the alpha channel on a pixel-by-pixel basis, but rather per object. Different parts of the object would have different levels of transparency depending on how much you wanted the background to show through. This allows you to create rectangular objects that appear as if they are irregular in shape -- you define the rectangular edges as transparent so that the background shows through. This is especially important for animation, where the background changes from one frame to the next.

Rendering overlapping objects that include an alpha value is called alpha blending.