Complex sinusoids are special inputs to LTI systems: the output has the same frequency, with only amplitude and phase changed.

For a continuous-time input

$$x(t)=e^{i\omega t},$$

an LTI system produces

$$y(t)=H(\omega)e^{i\omega t}.$$

For a discrete-time input

$$x[n]=e^{i\omega n},$$

the corresponding output is

$$y[n]=H(\omega)e^{i\omega n}.$$

The complex factor $H(\omega)$ is the frequency response at that frequency.

Its magnitude

$$|H(\omega)|$$

describes the gain, while

$$\arg H(\omega)$$

describes the phase shift.

The exact way $H(\omega)$ is computed depends on the representation of the system: from an impulse response, a Fourier representation, a transfer function or another equivalent model.

Frequency response separates a complicated input into frequency components and asks how the same system modifies each component.