The bessel is very poor in terms of sharp cutoff at a given frequency. However it is excellent at having a group delay that is near constant over the filter bandwidth, it gets better with higher orders., but the sharpness of the initial attenuation does not.
These filters are more often used as delay lines, a use where they are excellent.
The Butterworth Filter is the maximally flat amplitude filter. It provides a near 0 attenuation until near the cutoff frequency and then descends into attenuation smoothly. The tranition becomes sharper with higher orders. It has moderate group delay so it has some overshoot on sharp rising waveforms. this gets worse with higher orders.
The Chebyshev filter trades off flatness in the pass band for a steeper decline into the stop
band. You design a cheby with a recurring wavelike ripple of attenuation in the paasband of usually 0.05db to 3db. In return you get a much steeper portion of the attenuation curve near the cutoff frequency. waveforms are distorted by group delay errors more severely than in the butterworth. The higher the ripple the worse the distortion
The elliptical filter is like cheby^2 since it has ripples in the stopband and the passband.
It has been proven to be the fastest possible descent into the stopband.
The time delay errors are more severe than the cheby.
For all of these filters. when at a distance from the cutoff frequency they will attenuate 20db/decade in frequency times the filter order.