Hi Agisthos,
A few bullet points:
1st Order: gentle slope of 6dB/octave on all filters, that is, half the amplitude gets through at double the crossover frequency. Theoretically the best because it does not much phase shift the signal, giving very good amplitude and transient response. Disadvantage: considerable out of band energy passed to the drivers, so needs VERY good drivers which are expensive. Difficult to handle the out of band driver output, since it mixes with the in band signal from a very different driver, causing phase shift and lobing and beat problems.
2nd Order: stronger 12dB/octave slope on all filters, that is, one quarter the amplitude gets through at double the crossover frequency. However, there is a 180 degree phase inversion, which means drivers must be reverse wired, no big deal. Quite a good approach, but criticised by some for poor imaging performance.
3rd Order: vigorous 18dB/octave slope on all filters, that is, one eighth the amplitude gets through at double the crossover frequency. Widely used, and probably the most popular arrangement since out of band frequencies are minimised without too much cost on transient performance. No unhappy phase isues, and often praised for good imaging. Can use inexpensive drivers. Dimensioning a good 3rd order is quite tricky.
4th Order: Very strong 24dB/octave slope on all filters, that is, one sixteenth the amplitude gets through at double the crossover frequency. Rarely used on passive speakers (exceptions are the Ellis 1801B, and some of the B&W speakers) but widely used on active systems, such as the Orion and most pro-audio sound reinforcement systems. Excellent for high power, driver rating can be fully exploited because few out of band frequencies are passed on. No phase issues, but transient response can be compromised, and if configured for good transient response, ringing can be an issue. Multiple, series connected filters are complex mathematically and the effect with audio is to slightly dull the impact and slam. Some say detail is also compromised with the fourth order; I doubt this.
The order of the filter is a simple choice. The dimensioning of the filter, however, is problematic, and not even good software can make the design of a premium crossover an easy exercise. Further, the quality of the filter components is important, too, with inferior caps in particular highly audible.
Hope this helps,
Cheers,
Hugh