What does Construction and Standard/Alternate within the PCB Fab Industry?
What is the A in Constructions (AxB) and why would anybody care ?


B tells you the specific weave of glass being used.

A tells you how many sheets of woven glass are used to build that laminate layer.


  What does the Standard/Alternate column mean ?


Notice in your chart that lines 1 and 2 are both for a 0.002 inch thick layer. 

The Standard/Alternate column tells you that the 1x106 construction is the standard way to achieve this thickness, and the 1x1067 construction is an available alternate way to do it.

As the other answers have said, you might choose to use the alternate construction if the slightly different dielectric constant and dissipation factor suit your design better.