Below is a general description of the fine structure constant meant for a reader without a heavy background in science and math. For a more rigorous treatment, visit the wikipedia description.
What is the Fine Structure Constant?
The Fine Structure Constant is a special value in nature that describes how strongly a charged particle interacts with an electromagnetic field. It has a value of roughly 1/137 and scientists often refer to the constant simply as ?, the greek letter alpha.
Why is the Fine Structure Constant special?
Few numbers in science cannot be predicted theoretically (that is, they aren't directly produced by a theory such as the standard model) and the Fine Structure Constant is one of these such numbers. It can only be produced as the result of experimentation. It is a dimensionless number, which means it has no units associated with it.
The Fine Structure Constant is also a mystery. Since it is not produced by any theory and can only be determined experimentally, it has no definite origin. One can say that a number such as the mass of a proton comes from the quarks that make up a proton or the energy inside a proton but no such reasoning exists for the Fine Structure Constant. It is simply how things are.
Mentions of the Fine Structure Constant
The Fine Structure Constant is a popular topic among scientists and although I could never list every mention on the internet, I can point you towards some of the more interesting mentions and discussions.