The alpha value is used in the RGBA colour space to indicate the transparency of a colour. The alpha value goes from 0 to 1 where 0 is completely transparent and 1 not transparent at all.
This allows to do "alpha compositing" which, in lay terms, is the process of placing an image on top of a background and to combine the colours to create the illusion of transparency. For more information on this colour space, you can Google "rgba color space" or visit this Wikipedia page
An RGBA colour is usually denoted with 4 values, 3 for each of the RGB components and a last one for the alpha value (e.g. (0,0,0,0.5)), although in the Colour Style document you have referenced they are denoted with a % value in front of an hexadecimal RGB value (e.g. 50% (#000000)).
An "alpha value of black", at least in the scope of the referenced document, would be a short way to indicate any of the colours in a the RGBA space that have black as their RGB value, (0,0,0). They range from from 100% solid black (0,0,0,1) to completely translucent black (0,0,0,0) passing through all the possible values of transparency.
So, for example a 57% alpha value of black, denoted as 57% (#000000), would be an "alpha value of black" where the alpha is equal to 0.57 meaning it is 57% opaque.
In this image you can see the "alpha values of black" suggested in the Colour Style document. I have placed them on top of white and red so you can see the "alpha compositing" in action.