Define passband, stopband, αmax, αmin, ωp, ωs

Attenuation is $\alpha(\omega) = -20\log_{10}|T(j\omega)|$ dB. The figure shows the specification of a low-pass filter:

 alpha(dB)
     |                 :  ////////////////
amin |- - - - - - - - -:-+    stopband
     |    passband     : |
     |                 :/  <- transition
amax |=================+
   0 +-----------------+--+---------------> w
                       wp ws

• Passband: the band of frequencies the filter passes with little attenuation, $\alpha \le \alpha_{max}$. For a LPF it is $0 \le \omega \le \omega_p$.
• Stopband: the band the filter rejects, with $\alpha \ge \alpha_{min}$. For a LPF it is $\omega \ge \omega_s$.
• $\alpha_{max}$: the maximum attenuation allowed in the passband (passband ripple), e.g. 0.5 dB or 1 dB.
• $\alpha_{min}$: the minimum attenuation that must be achieved in the stopband, e.g. 40 dB.
• $\omega_p$: passband edge frequency, the highest frequency (for LPF) where attenuation is still within $\alpha_{max}$.
• $\omega_s$: stopband edge frequency, from where attenuation is at least $\alpha_{min}$.
• Bandwidth: the width of the passband. For a LPF it is $\omega_p$ (or $\omega_{3dB}$ for half-power bandwidth). For a BPF it is $BW = \omega_2 - \omega_1$, with centre frequency $\omega_o = \sqrt{\omega_1\omega_2}$.

 |T|   band-pass
  1 +      .-----.
    |     /       \
    |    /         \
    +---+---+---+---+---> w
       w1   wo     w2
        <--- BW --->

The region between $\omega_p$ and $\omega_s$ is the transition band; the ratio $\omega_s/\omega_p$ (selectivity) together with $\alpha_{max}$ and $\alpha_{min}$ fixes the filter order.