The electrical circuit (which may be as simple as a single capacitor or a complex network of inductors, capacitors and resistors) that connects the amplifier to a loudspeaker driver (or multiples of the same driver) is correctly called a filter. In a passive loudspeaker system, the electrical properties of the filter works in conjunction with the impedance of the driver to which it is connected to effect the change in amplitude response of the driver. In the case of a tweeter, the filter (which is called a high-pass filter) attenuates the signal getting through to the tweeter as frequency drops. The higher the order the filter, the more steeply the signal gets cut. A first order electrical filter cuts at 6dB per octave, a 4th order at 24dB per octave.
A crossover is the outcome when you apply complementary filters to a pair of drivers, e.g. low-pass to midwoofer and high-pass to tweeter, so that the net acoustic result is a seamless and flat 'frequency response' over the crossover range. This, unfortunately, is easier said than done.
Why? Three simple reasons.
First, it is a acoustic result that matters. In other words, the 'order' of the electrical filter does not translate directly to the 'order' of the crossover result. This is because loudspeaker drivers do not have flat frequency response outside their intended operating range. For most tweeters, you will see that the acoustic response drops from around 2kHz and below. Unless the high-pass filter is applied at least two octaves (i.e. at 8kHz) above that range, the natural roll-off of the tweeter will add to the 'order' of the crossover. So a simple capacitor, which is a first order electrical filter, combined with a rolling-off tweeter at 2kHz, is anything but a first order crossover. More likely a second order or third. The lesson from this is actual crossover order = electrical filter order + natural driver roll-off order.
Second, a passive filter works in conjunction with the impedance of the driver to which they are connected. That impedance is rarely flat, which is what all crossover formulas assume. This means that the electrical filter that promised you a second order electrical roll-off will deliver something a little less predictable. If the driver, particularly tweeters and midrange drivers, has a huge impedance spike within the crossover range, then all bets are off. Unless you can implement another circuit to flatten that impedance first. The lesson from this is textbook filters rarely work as intended, especially in the passive filter domain. A properly designed filter must take into consideration the natural acoustic response of the driver and its impedance characteristics. This is almost impossible without measured data and extensive modelling. Computer software such as LspCAD helps significantly.
Third, acoustic phase is problematic. That's the bad news. The good news is that phase is directly linked to frequency response and that can be 'predicted' with good modelling tools. Why is phase a problem? In a worst-case scenario, the resulting output from the drivers cancel each other out over some or all of the overlap because they are at 180-degrees to each other. This results in a crappy sound and very little music. The idea for phase alignment is so that the outputs of the driver pair in question sum flat over the crossover range. For odd-order crossovers, the phase is 90-degrees out of phase, but the acoustic response of a properly designed loudspeaker should still sum flat. There are well established crossover 'types' such as Linkwitz-Riley, Butterworth, Bessel, Chebychev etc., which trades amplitude response vs power response vs phase alignment depending on what's more important to you. Oh, the other thing is that actual phase alignment also depends on the relative position of the driver pairs. All of the crossover theory assume that the drivers are acoustically aligned, which is NEVER the case for a loudspeaker with a vertical baffle and flush-mounted drivers. However, these can be compensated for in the crossover, but you will need actual measurements and modelling tools. The lesson from this is loudspeaker design is more miss than hit, unless you have actual* measurement data.
Hope that helps.
James
* published graphs from manufacturers don't count because these are normally measured on an IEC baffle or infinite baffle, which your loudspeakers are unlikely to mimic. The size and shape of baffles affect the acoustic response of drivers.