The discrete-time Fourier transform of the system impulse response $h[n]$ is known as the frequency response of the system $H(\omega)$. The frequency response defines the response of the system to a single frequency. That is, if $x[n] = e^{j \omega_0 n}$, then $$ \begin{align} y[n] &= H(\omega_0) x[n] \\ &= H(\omega_0) e^{j \omega_0 n} \; . \end{align} $$ If we take the DTFT of both sides, we get $$ \begin{align} Y(\omega) &= H(\omega_0) \delta(\omega - \omega_0) \; . \end{align} $$ This shows from a different perspective that a single frequency input yields a single frequency output.
One of the most common representations of frequency domain signals is the magnitude-phase form. In this representation, the frequency information equals (for the Fourier series, continuous-time Fourier transform, and discrete-time Fourier transform) $$ \begin{align} c_k &= |c_k| e^{j \left( \angle c_k \right)} \\ X(\Omega) &= |X(\Omega)| e^{j \left( \angle X(\Omega) \right)} \\ X(\omega) &= |X(\omega)| e^{j \left( \angle X(\omega) \right)} \\ \end{align} $$ The magnitude describes the strength of each frequency in the signal. The phase describes the sine/cosine phase of each frequency. The phase can also be thought of as the relative proportion of sines and cosines in the signal (i.e., a phase of zero contains only cosines and a phase of 90 degrees contains only sines). Phase generally relates to the delay of the signal.
The magnitude response of a frequency-domain signal is defined by $$|X(\omega)| = \left( \textrm{Re}\{ X(\omega) \}^2 + \textrm{Im}\{ X(\omega) \}^2 \right)^{1/2} \; ,$$ where the $\textrm{Re}\{ X(\omega) \}$ represents the real part of $X(\omega)$ and $\textrm{Im}\{ X(\omega) \}$ represents the imaginary part of $X(\omega)$.
The phase response of a frequency-domain signal $X(\omega)$ is defined by $$\angle X(\omega) = \tan^{-1} \left[ \frac{\textrm{Im}\{ X(\omega) \}}{\textrm{Re}\{ X(\omega) \}} \right] \; ,$$ where the $\textrm{Re}\{ X(\omega) \}$ represents the real part of $X(\omega)$ and $\textrm{Im}\{ X(\omega) \}$ represents the imaginary part of $X(\omega)$.
For the system with frequency response $H(\omega)$, frequency-domain input $X(\omega)$, and frequency-domain output $Y(\omega)$, we can represent the signals by $$ \begin{align} Y(\omega) &= H(\omega) X(\omega) \\ |Y(\omega)| e^{j \angle Y(\omega)} &= |H(\omega)| e^{j \angle H(\omega)} |X(\omega)| e^{j \angle X(\omega)} \\ |Y(\omega)| e^{j \angle Y(\omega)} &= |H(\omega)| |X(\omega)| e^{j (\angle H(\omega) + \angle X(\omega))} \\ \end{align} $$
Hence, The magnitude of the frequency-domain output of a system can be represented by the system magnitude response multiplied by the magnitude of the frequency-domain input, $$ \begin{align} |Y(\omega)| = |H(\omega)| |X(\omega)| \end{align} $$ The phase of the frequency-domain output of a system can be represented by the system phase response added to the phase of the frequency-domain input, $$\angle Y(\omega) = \angle H(\omega) + \angle X(\omega)$$
The phase delay of a system $$ \begin{align} \tau_{ph}(\omega) = - \, \frac{\angle H(\omega)}{\omega} \end{align} $$ represents the delay on a single frequency input into the system.
The group delay of a system $$ \begin{align} \tau_{gr}(\omega) = - \, \frac{\partial}{\partial \omega} \angle H(\omega) \end{align} $$ represents the delay of a group of frequencies around $\omega$. Note that the phase delay and group delay are equal when the phase is linear.