hiya,
a crossover comes in 2 main forms, mechanical and electronic,
they basically separate the sound into hi freq, mid freq and Bass freq and send it to the appropriate speaker,

(no point sending 20khz to a sub that cuts off at 100hz)

with your crossovers configured properly you can squeeze every last watt out of your speakers without distortion.

iv'e just done an ice install in my mr2 and it seems the x-overs are built into the factory amps

(please correct if wrong)

i highly reccomennd either getting a head unit with adjustable crossovers built in or even better is an amplifier with adjustable xovers.

if you wish to run component speakers

(separate woofers and tweeters) a decent pair will come with a crossover box, even though it has a standalone xover it is still very important to use either the head unit or the amps xover to limit the frequency sent to the speakers to achieve maximum efficiency.

say your sending full range

(15hz-24khz appx) to a speaker,
the average person can hear about 30hz to 18khz
so even before we've got into what the speaker is capable of

we can see that there is no point in a speaker emitting 15hz-30hz and 18khz-24khz because we can't hear it and its just wasting power, the same applies to the range of the actual speaker,

this is how i have mine setup

sub

- 30hz

->150hz
6x9

- 100hz

->18khz

(has own x-over for tweeters)

doors

- woofers 100hz-10k

- tweeters 8.5k

- 18k

-3db

hope this helps
Ta Marc