For an SISO linear system, it’s transfer function is $T(s)$ such that
$y=T(s)u$
What essentially does the transfer function mean? It represents nothing but the ratio and phase offset between output and input under different frequencies. By substituting $s$ with $j\omega$ to the transfer function $T(s)$, we can get the ratio and phase offset. It should be noted that transfer function only describe the relationship between input and output; it is only applicable to linear systems. The analysis of properties of linear systems based on transfer functions form the classic frequency analysis.