Classifying Discrete time filter
It's a bandpass filter whose frequency response suppresses the low and high frequencies (with its zeros at $\omega = 0$ and $\omega = \pi$) and amplifies the band in between.

The bandwidth of the filter can be defined by reference to $-3_{dB}$ points, which would give you approximately $W_{-3dB} \approx 0.2\pi$, with a center frequency of $\omega_c = 0.5\pi$.

I would however not consider this bandwidth to be a descriptive one.