the transfer function
If a system is a machine that takes an input and produces an output, the transfer function is the single object that captures, completely, what that machine does — measured in the s-domain. It is the ratio of output to input once you have translated both into Laplace transforms and assumed the system starts from rest. One function H(s) tells you everything about the system's behaviour.
Start from a linear constant-coefficient ODE such as a x'' + b x' + c x = f(t) with zero initial conditions. Transform both sides: (a s^2 + b s + c) X(s) = F(s). Solving for the ratio of output transform to input transform gives H(s) = X(s)/F(s) = 1/(a s^2 + b s + c). The denominator is the system's characteristic polynomial — the same polynomial whose roots gave the natural modes back in the homogeneous theory. So the transfer function is literally the reciprocal of the characteristic polynomial (times any numerator the input coupling adds), and to find the output for a given input you simply compute X(s) = H(s) F(s) and invert.
The transfer function is the central object of linear systems, signals, and control. Its inverse transform is the impulse response h(t), so 'multiply by H(s)' in the s-domain is the same as 'convolve with h(t)' in time. Its denominator's roots, the poles, dictate stability and the natural frequencies; its behaviour as s travels along the imaginary axis gives the frequency response. Nearly every question about a linear system becomes a question about H(s).
For x'' + 3 x' + 2 x = f(t) the transfer function is H(s) = 1/(s^2 + 3s + 2) = 1/((s + 1)(s + 2)); given any input F(s), the output is X(s) = F(s)/((s + 1)(s + 2)).
Output transform = H(s) times input transform; the factored denominator shows the poles at -1 and -2.
The transfer function is defined for ZERO initial conditions; it describes the system's response to the input alone. Nonzero initial conditions add separate terms (the system's free response) that the transfer function does not include.