Note: this article and video were created when Lightroom 5 came out, but they are applicable to later versions as well.
Lightroom smart previews allow you to develop your photos, even when the masters are offline. Imagine traveling with your laptop, and leaving your big heavy external hard drive with all your photos back at home. Work on your photos on the road, and when you get home and reconnect your hard drive, your work is seamlessly applied to your masters. All you need on the road are your laptop your catalog, and smart previews of your photos. Not interested in traveling with your photos? Smart previews may still give you a performance benefit in the Develop module. Finally, Adobe has hinted that at some point in the future, smart previews will enable us to work with our photos in a Lightroom app on our mobile devices.
Watch this video for more on how smart previews work, and how to build and manage them. This is one video of 58 in my Lightroom 5: The Fundamentals & Beyond series (which as of 2016 has been updated to: Lightroom CC/6 and 5: The Fundamentals & Beyond and now contains 76 videos.)
(For higher quality, once you hit Play, click on the sprocket wheel in the bottom right and choose 720/HD.)