How frequency response related to a transfer function
An LTI system's "frequency response" tells you how the system acts on the amplitude and phase of a sinusoidal input. If the frequency response is $H(f)$, then an input $x(t)=e^{j2\pi f_0t}$ produces an output $y(t)=|H(f_0)|e^{j(2\pi f_0t+\angle H(f_0))}$. It is common to divide the frequency response in two, the gain $|H(f)|$ and the phase $\angle H(f)$.

The same system's "transfer function" is defined as follows: if an input $x(t)$ produces an output $y(t)$, then the system's transfer function is $H(s)=\frac{Y(s)}{X(s)}$, where $X(s)$ and $Y(s)$ are the Laplace transforms of $x(t)$ and $y(t)$.

When the transfer function is evaluated at $s=j\omega$, it is also known as the systems's frequency response. Note that the transfer function is more general than the frequency response, and can provide more insight into a system's behavior, for example about transient response or stability.

These definitions can also be extended to non-linear systems but that is beyond my experience.