A first order crossover uses a single filter (inductor and capacitor) to separate frequencies into low and high frequencies and then direct them to the woofer and tweeter.
A second order crossover uses a double filter. A third order crossover uses a triple filter, etc. The higher up you go, complexity increases.
Also, the higher the order, the more sharply that the frequency rolls off in this process of frequency filtration. A first order crossover rolls off at 6dB per octave, a second order at 12 dB, a third order at 18dB.
There are numerous advantages and disavantages to each. The cross-over is not chosen in isolation. In choosing a cross-over, the designer has to have regard to driver and cabinet characteristics. In other words, it's a package. The designer makes trade-offs. Disadvantages in a cross-over can be compensated for in cabinet design and driver choice, and vice-versa. For example, a first order cross-over, because of gradual frequency roll-off, will have a larger band of frequencies shared by both the woofer and tweeter. This can produce phase errors where the same frequency comes out of both drivers at different points in time. So, for example, the designer might compensate by placing the front plane of the tweeter slightly behind the plane of the woofer so that when the same frequency comes out of both drivers, one travels a longer distance to reach you This compensates for the phase error otherwise produced by the cross-over. Another designer might prefer to tackle phase error by using a higher order crossover, where there is less overlap of frequencies because of sharper frequency roll-off. But he would then have to deal with other problems unique to the higher order cross-over. I don't have sufficient technical knowledge to describe all of the advantages and disadvantages of each. The bottom line is that there is no one type which is best. Cross-over choice has to have regard to the drivers and cabinet characteristics so that they complement each other.