Transfer Function, what does s Stand for

Question

Trying to understand what the variable s is for transfer functions (if there is a common accepted use of it). In the problem space I am working in, control theory, I believe I have seen definitions of "short period poles", "complex Laplace transform variable", and at http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf it describes s as s = sigma + j * omega (I guess as a complex variable). Any insight into one of these being correct or if I am missing a better common use, that would be appreciated.