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The crossover simply sends a range of frequencies to a particular speaker. In a component speaker set up it will send the high frequencies to the smaller tweeter and the lower frequencies to the bigger mid driver. This helps to prevent distortion and loss of sound quality. The tweeter is only designed to handle a high freq. signal, anything lower with a longer wave would distort the tweeters sound. on the flip side the mid driver is designed to push the lower freq. and would not be able to replicate a cymbal's sound accurately.
The crossover is wired up to your amp or head units speaker leads and then it sends the appropriate signals to the appropriate speakers via two sets of speaker wires (one to the tweeter and one to the woofer/midbass driver).
They are called crossover networks because their frequency responses cross over (have the same attenuation at some audio frequency). A crossover network is really two networks, a low pass filter (passing only low audio frequencies) and a high pass filter (passing only high audio frequencies) so that the woofer receives only low audio frequencies and the tweeter receives only high audio frequencies.