The Smart Object has to be one of the more powerful tools in the Photoshop arsenal. If it still confuses you a little — and I know it does — check out this in-depth video on the myriad ways in which it can be utilized.
It took me a while to figure out how I could incorporate Smart Objects into my workflow. But, when I had a better grasp of how to use them, they sped up my workflow like no other tool I had learned to use up to that point. Their major and most obvious feature is that they are nondestructive, meaning that you cannot make any permanent changes to them. The simplest example of this, as Unmesh Dinda of PiXimperfect points out, is when you decrease the size of a Smart Object, no pixels are lost, so when you resize the image to its original dimensions, it looks exactly the same. If you try to do this with a regular/rasterized layer, the image will end up blurry, because data has been discarded during the decreasing. There are so many ways this comes in handy, especially for compositing.
The most frequent way in which I use Smart Objects is when I need to work on the original raw file in Photoshop. Instead of having to roundtrip from Photoshop to Lightroom or as an alternative to using the Camera Raw filter, I just open my images as Smart Objects from the start. As Dinda points out, it's as simple as double-clicking the Smart Object layer, and poof, Camera Raw pops up. I find this handiest when I'm blending layers in Photoshop and I haven't quite equalized the white balance between flash and ambient exposures.
These are just two small examples that have helped to make my post-processing more efficient. Many more are contained in the video. It's a long one, so grab a coffee and get comfortable.