An explanation of "phase" of a filter
The phase of a sinusoid $s(t)=A_0\cos(2\pi f_0 t + \phi_0)$ is $\phi_0$ radians.

If this sinusoid goes through an LTI system with frequency response $H(f)$, then the ouptut is $y(t) = |H(f_0)| A_0 \cos(2\pi f_0 t + \phi_0 + \angle H(f))$. So, the phase of the output is different than the phase of the input -- the system introduced a phase shift in the signal.

Note that you can interpret the phase shift as a time delay. If $T_0=1/f_0$, then a phase shift of $2\pi$ corresponds to a delay of $T_0$. This allows you to calculate that the delay $\Delta$ for a phase shift $\theta$ is $$\Delta = \frac{T_0 \theta}{2\pi}.$$ Note that the delay is a function of the sinusoid's frequency. An LTI system that produces the same time delay for all its inputs is said to have linear phase.

Note that LTI systems without linear phase will in general distort their input, even if they have constant gain; this is the reason we want to design linear phase systems. In the case of filters, usually we only require linear phase in the filter's passband.

(Note that this explanation is for continuous time, but the ideas are also valid for discrete time).