I can't beat the Wikipedia page on crossovers, so I'll just block quote it:
First Order
First-order filters have a 20 dB/decade (or 6 dB/octave) slope. All first-order filters have a Butterworth filter characteristic. First-order filters are considered by many audiophiles to be ideal for crossovers. This is because this filter type is 'transient perfect', meaning it passes both amplitude and phase unchanged across the range of interest. It also uses the fewest parts and has the lowest insertion loss (if passive).
Second Order
Second-order filters have a 40 dB/decade (or 12 dB/octave) slope. Second-order filters can have a Bessel, Linkwitz-Riley or Butterworth characteristic depending on design choices and the components used. This order is commonly used in passive crossovers as it offers a reasonable balance between complexity, response, and higher frequency driver protection. When designed with time aligned physical placement, these crossovers have a symmetrical polar response, as do all even order crossovers.
Third Order
Third-order filters have a 60 dB/decade (or 18 dB/octave) slope. These crossovers usually have Butterworth filter characteristics; phase response is very good, the level sum being flat and in phase quadrature, similar to a first order crossover. The polar response is asymmetric. In the original D'Appolito MTM arrangement, a symmetrical arrangement of drivers is used to create a symmetrical off-axis response when using third-order crossovers.
Fourth Order
Such steep-slope filters have greater problems with overshoot and ringing[3] but there are several key advantages, even in their passive form, such as the potential for a lower crossover point and increased power handling for tweeters, together with less overlap between drivers, dramatically reducing lobing, or other unwelcome off-axis effects. With less overlap between adjacent drivers, their location relative to each other becomes less critical and allows more latitude in speaker system cosmetics or (in car audio) practical installation constraints.
Of course I recommend you read the whole article and then do some more web searches. There is no one answer to your question, as you can see it depends very much on the situation and other components involved.
You also might find this section partularly interesting:
Active crossovers can be implemented digitally using a DSP chip or other microprocessor. They either use digital approximations to traditional analog circuits, known as IIR filters (Bessel, Butterworth, Linkwitz-Riley etc.), or they use Finite impulse response (FIR) filters. IIR filters have many similarities with analog filters and are relatively undemanding of CPU resources; FIR filters on the other hand usually have a higher order and therefore require more resources for similar characteristics. They can be designed and built so that they have a linear phase response, which is thought desirable by many involved in sound reproduction. There are drawbacks though—in order to achieve linear phase response, a longer delay time is incurred than would be necessary with an IIR or minimum phase FIR filters. IIR filters, which are by nature recursive have the drawback that if not carefully designed they may enter limit cycles resulting in non-linear distortion.
