What is a transfer function and what does it represent?

A transfer function G(s) is the ratio of the Laplace transform of output to input, assuming zero initial conditions. It completely characterizes a linear time-invariant (LTI) system's input-output behavior in the s-domain. The transfer function is independent of input signal and describes the system's inherent dynamics. Poles (denominator roots) determine stability and natural response modes; zeros (numerator roots) affect the response shape.