As we mentioned above, the butterworth filter is monotonic in both the passband (maximally flat) and the stopband (keeps rolling off forever), but it's rolloff is somewhat slow. I.e., frequencies that close to the cutoff frequency aren't attenuated too much. The Chebyshev filters allows us get a faster rolloff, in exchange for sacrificing the monotone property.
For a type I Chebyshev filter, we trade passband flatness for a faster rolloff. In the example Chebyshev I filter in figure 2, we allowed 2 dB of ripple in the passband (i.e. frequencies below the cutoff could be anywhere from -2 dB to 0 dB). In exchange, we get a monotone stopband that is a little sharper than a butterworth (note that all of the filters in the example have the same -3 dB point). If we had allowed more ripple in the passband, we could get a sharper stopband. For a type II Chebyshev filter, the passband is maximally flat, but now the stopband doesn't keep rolling off forever, it stops at some point. In the example, we've specified 20 dB attenuation and gotten a much sharper rolloff.
We also need to point out that for most of the filters, when we specify the cutoff frequency, that means the frequency at which the response is attenuated to 0.707 of the input (i.e. the half power point, aka the -3 dB point). For the butterworth, this is exact, and for the the Chebyshev type I its approximate (depending on how much ripple we specified). But for the Chebyshev type II, the cutoff frequency is the frequency at which the specified stop band attenuation is first reached. You can see this by comparing left and right plots in 2. In the left figure, the Chebyshev II filter need a 23.4 Hz cutoff frequency in order to match the other filters -3 dB point at 10 Hz.
