Filters can get complicated (messy) quickly but fundamentally it is how you define the poles and zeros in the transfer function that determine if it is Butterworth, Chebyshev, elliptical, etc. Butterworth filters have no ripple in the passband or the stopband. Elliptical filters have ripple in the passband and stopband but provide steep rolloffs and less attenuation at the passband edge. Chebyshev puts ripples in the passband or stopband, not both, and you decide which band is appropriate for your application. Bessel filters are linear phase, meaning constant group delay, which means better pulse fidelity (impulse/time-domain response) than other types. Bessel and its variants provide linear phase without ripples but have low roll-off. And so forth and so on...
This Wikipedia link has some pictures that show what I am babbling about:
https://en.wikipedia.org/wiki/Filter_(signal_processing)
Perhaps someday I could write an overview but I am working far too many hours lately (did the simulation whilst awaiting test results, in the lab, on Sunday, fourth in a row!) And I'd have to find my old grad textbooks, and I didn't really like the last one that much...
And frankly I am waiting for the drama to die down.
Speaker crossovers use passive LC circuits as opus112 said so they can handle the high power. I have long espoused that an active design provides better overall sound but there is a lot of resistance to that idea. These days DSP provides excellent filter response but of course you must have a DAC in the circuit (and an ADC, someplace).
By the way, the simulation I showed did not use work tools (get in trouble for that), or even my personal simulator (for which I paid good money), but a free program called LTSpice available from the Linear Technology website. Anyone is welcome to download and play with it, using my files or whatever they choose.