What is the significance of 137?

It is the inverse of the Fine Structure Constant, 1/137. 

In quantum electrodynamics (QED), the study of electromagnetic (light waves) interaction, there is a constant that physicist Richard Feynman speaks of that is regarded as "pretty cool" in physics.  If you take the charge on an electron squared, divide it by the product of planck's constant and the speed of light, you get a number that is very close to 1/137.  It is apparently a significant number in QED.

e^2/(hc) = 1/137, in a certain unit system.

It has been joked by Feynman that if a physicsist's car breaks down on a desert road, another physicist will stop to rescue you if you scribble "137" on the "help needed sign."

I'll have to look into the actual significance of the fine structure constant, as I do not know what it really means yet.