The transfer function of a system is the Laplace transform of the impluse response of the system. It is usually expressed as a ratio between the output and input to the system. It can be used to determine physical characteristics of the system such as stability and frequency response.

G(s) = Y(s)/R(s)

E.g.
For a simple RC network we may write:

V₁(s) = (R+(I/Cs))I(s) and
V₂(s) = I(s)/Cs that results

G(s) = V₂(s)/V₁(s) = 1/(RCs + 1)

where RC is a time constant and
G(S) has a single pole s = -1/RC

The transfer function (or impulse response) of a linear system is the complete characterization of the system. Once it is known, the output to the system may be determined by two methods:

Method 1: Multiplying the transformed impulse response and inputs and taking the inverse Laplace transform to yield the time-domain output.

Method 2: Performing the time-domain convolution of the inverse Laplace transformed transfer function and the input waveform.

Typically Method 1 is the preferred method, as many EE students studying introductory Linear Systems will attest to.