Why An elliptic (Cauer) filter trades ripple in both the passband and stopband for the fastest possible transition from pass to stop. Those stopband transmission zeros show up as deep notches just above (or below) the cutoff frequency, which is what makes the cutoff so steep for a given number of poles. It is the filter of choice when you must reject a signal very close in frequency to one you want to keep.
Watch out The description of an extremely flat passband with gradually rounded stopband corners is the Butterworth (maximally flat) response, and the gradual-rolloff answers do not describe any sharp-cutoff design at all.
Elliptic = Extreme edge plus notches. Butterworth = flat and Bland, Chebyshev = ripple in the passband.