While a loudspeaker driver may be excellent at what it's designed to do, very few speaker drivers can cover the full range of human hearing from 20hz to 20khz. As a result, multiple drivers are almost always used in speakers; some for high frequencies, and some for low. There are endless permutations on these; some speakers use a tweeter that can handle the midrange, and hence "cross over" to the woofer at a low frequencies, while others apply a midrange driver to cover the area between where a large woofer and a tweeter can perform.
Of course, you don't want low-frequency signals sent to the tweeter, nor high-frequency to the tweeter - in this way, you can have them produce sound in the areas where they distort the least. This device, which determines where the signal "crosses over" from a low-frequency driver to a high-frequency driver, is hence referred to as a "crossover".
Because even the best speaker drivers work better in some frequencies and worse in others, a crossover can make or break a speaker. There are numerous examples of half-cocked DIY projects using $160 Scan-Speak tweeters and $300 Eton woofers that sound awful because the designer chose a crossover ill-suited to the components, and speakers using $17 woofers and $9 tweeters that sound marvellous because the designer was able to use the best parts of both drivers. In addition, a careful crossover design can help correct for artifacts in frequency response - dips and peaks can be neutralized.
Most crossovers are passive devices, having no components that require any power. These usually consist of capacitors and inductors in various configurations of series and parallel to determine functionality.
While the electrical engineering behind these devices is somewhat complex, the simplified version of it is as follows: Capacitors will reduce power moving through them 75% every time the frequency is drecreased by 50%, or, in other words, they reduce volume from the speaker driver by six decibels every time the frequency is moved down an octave. Inductors work much the same way, except backwards, reducing power moving through them 75% every time the frequency is decreased by 33% (6 decibels per octave down). Past a given point, the decrease in power is so tiny it's almost immeasurable - hence, capacitors are useful for sending high frequencies to tweeters, and inductors for sending low frequencies to woofers.
Inductors are measured in Henries or Micro-Henries (uH), while capacitors are measured in Farads or Micro-Farads (uF). The value of these components determines what frequencies they will let through, and what they will not - a larger inductor's effects are noticeable at increasingly low frequencies, while a smaller capacitor will attenuate the frequency at much greater levels. In other words, bigger inductors block lower frequencies, while bigger capacitors let through lower frequencies.
However, because two capacitors or inductors wired in series will simply act as a single inductor or capacitor, they are usually combined: For example, a capacitor may be used in series with a tweeter, followed by an inductor in parallel. The capacitor blocks low frequencies, and what is not blocked by the tweeter will then bypass the tweeter through the inductor. As a result, 12db per octave is lost. The number of components corresponds to the resulting drop per octave; a two-component crossover as above is called a "second-order" crossover with a 12db/octave drop, while one with an additional capacitor after the inductor would be a "third-order" crossover with a 18db/octave drop.
In addition, capacitors and inductors can be used to fix problems in frequency response. For example, the Bohlender-Graebner Neo3PDR tweeter, which I am using in a project, has a frequency response peak around 3khz. As a result, most people put in parallel with the tweeter an inductor and capacitor : The capacitor allows through frequencies from about 11khz up, the inductor allows through frequencies below 9khz. Between these two points, the frequency is decreased by 6db/octave up from 9khz and down from 11khz - just the right amount to eliminate the peak.
I should stress that crossover design is tricky, and I, as a newbie myself, strongly recommend that you consult someone more experienced than yourself - or, better still, use a tried-and-true design, such as I provided links for yesterday. Designing a crossover by hand is very difficult, and while most good speaker design programs include excellent crossover design functionality, these tend to be both expensive and difficult to use.