It's not quite as complex as the first post states.
Let's say you have a two way system and you want the crossover point to be 1000 Hz. The crossover separates the frequencies so each driver gets the appropriate ones, below 1000 to the woofer, above 1000 to the tweeter. However, there is no such thing as a perfect crossover so it can't just stop at 1000, (termed a brick wall filter) which in the case of the woofer would be 999 getting through at full strength, but 1001 being completely blocked.
The order tells you how fast the response drops off beyond the cutoff frequency like loontoon says and this is determined by the complexity of the crossover as Trejla points out. The higher the order the faster the signal drops off beyond the cutoff frequency, i.e. the more it acts like a perfect brick wall type filter. So it seems that higher order would be better, but all filters introduce phases shifts, and the higher the order the more complex these phase shifts become, and depending on the design it can also start to have amplitude peaks and dips in it's response that extend into the frequency range it is designed to pass.
The fourth order Linkwitz-Riley that you asked about is a design that some feel has a roll off rate and phase response that is the best compromise between a higher rolloff and flatter phase response. Others feel that anything beyond a simple first order filter introduces too much phase shift to be acceptable.
A full understanding does require a lot of study and a lot of math but the basics are just that, pretty basic.
Let's say you have a two way system and you want the crossover point to be 1000 Hz. The crossover separates the frequencies so each driver gets the appropriate ones, below 1000 to the woofer, above 1000 to the tweeter. However, there is no such thing as a perfect crossover so it can't just stop at 1000, (termed a brick wall filter) which in the case of the woofer would be 999 getting through at full strength, but 1001 being completely blocked.
The order tells you how fast the response drops off beyond the cutoff frequency like loontoon says and this is determined by the complexity of the crossover as Trejla points out. The higher the order the faster the signal drops off beyond the cutoff frequency, i.e. the more it acts like a perfect brick wall type filter. So it seems that higher order would be better, but all filters introduce phases shifts, and the higher the order the more complex these phase shifts become, and depending on the design it can also start to have amplitude peaks and dips in it's response that extend into the frequency range it is designed to pass.
The fourth order Linkwitz-Riley that you asked about is a design that some feel has a roll off rate and phase response that is the best compromise between a higher rolloff and flatter phase response. Others feel that anything beyond a simple first order filter introduces too much phase shift to be acceptable.
A full understanding does require a lot of study and a lot of math but the basics are just that, pretty basic.