The fine structure constant, alpha, has no units of measure, so it is like a ratio. PI is another ratio of the diameter of a circle to its circumference. It is very much related to the charge amplitude of an electron.

Wikki has a good article on it.

The fine structure constant is the ratio between the velocity of the electron in the Bohr model of the atom and the speed of light. The square of alpha; is the ratio between the electron rest mass (511 keV) and the Hartree energy (27.2 eV = 2 Ry).

# In the theory of quantum electrodynamics, the fine-structure constant is the coupling constant for the strength of the interaction between electrons and photons. The theory does not predict its value; thus it must be determined experimentally. In fact, it is one of the 20-odd 'external' parameters in the Standard Model of particle physics.