2nd curtain flash, for those unfamiliar with the term, is having the flash fire at the end of the exposure rather than at the beginning. Normally you want the flash to fire as soon as you press the button. When shooting something moving, you want it to fire at the end.
If you shoot it at the beginning, you capture the object with the flash and then the moving part of the image. That's the opposite of the way that you want the picture to look. The story you want the picture to tell is about how the object got where it is rather than where it went after you shot it.
The term "2nd curtain" derives from the fact that the mechanical shutter used in SLRs and DSLRs is really a pair of curtains. The first drops revealing the image to the film/sensor and the second comes down covering it up again.
When shooting a particularly high shutter speed picture, the second curtain starts moving down before the first curtain has reached the bottom. The image is captured through a rapidly moving opening between the two curtains. That's why most flashes only work up to 1/125 or 1/250 shutter speeds. The flash is nearly instantaneous, so it would only illuminate the strip of the image that is exposed between the two moving curtains when it fires. Flashes with high speed sync modes work by extending the duration of the flash. It's not as bright but it lasts for the entire exposure. It seems counter-intuitive to think that a high speed flash sync relies on a longer duration flash, but that's the way it works.