The name Chebyshev is used because the design involves the eponymous polynomial as an approximation method.
The ideal filter response is 1.0 gain in the passband, 0.0 (or a certain maximum amount) in the stop band, and going from one to the other over a specified transition range.
Since an ideal square (brick wall) filter requires infinite poles, is either noncausal (starts moving before a signal is applied) or exhibits infinite time delay (but doesn't start moving before the signal is applied), and has poor step response (it exhibits both a sin x/x impulse response and Gibbs phenomenon), it's not very practical.
So, we make the sacrifice that the transition band must be continuous and gradual, and we have many degrees of freedom to traverse than "1 to 0" band.
The tightest approximation, in absolute terms, is the Butterworth. Which, in more abstract mathematical terms, is an orthogonal series of polynomials, of increasing order, with some characteristic property. The polynomials arise in fitting a polynomial curve to this problem; the metric is minimum RMS error, thus, the frequency response is maximally flat.
Chebyshev approximations work within a peak error band, rather than a mean or RMS weighted error metric. This is why Chebyshev filters are specified with pass/stop band ripple. The limit as ripple --> 0 is a Butterworth filter.
If, instead of approximating frequency and amplitude response exclusively, you dig deeper and find the phase shift / group delay, you can run some numbers on those. If you go for maximally linear phase, you get a Bessel filter.
All all-pole filters have ripple (or not) in the passband and an asymptotic drop in the stop band. You can't add nulls in the stop band, because those are zeros...
Filters with zeros (L||C links or L+C branches) have ripple in the stop band. The passband response can still be any of the traditional types, but by adding zeros, the same can be done for the stop band as well (in reciprocal, of course).
The Cauer / Elliptical filter does this, using all poles and all zeros (Nz = Np) to achieve the sharpest (most sudden) possible frequency cutoff, at complete cost to asymptotic attenuation (it's flat at -xx dB, not a slope of -xx dB/dec), (usually?) having a Chebyshev response in the passband.
Cheb. filters are generally used where sharpest frequency response is required: analog radio and audio applications, for example. Sound card outputs are an excellent example, typically having horrible step response -- but a frequency response of just over 20kHz for a sample rate of 44.1kHz, or whatever. (Or these days, since digital is so cheap, it's typically upsampled and sinc interpolated, thus pre-filtering it in digital to get by with an even cheaper analog filter.) They are avoided where excellent time-domain response is necessary: scopes*, digital radio, etc.
*Except some of the, DPO3000 or something like that Tek scopes were "100MHz" with lots of overshoot, another sign of their corporate change..
Tim