Older versions of the device used a Bluetooth 2.0 module which meant the accompanying app (iOS and Android) took forever to connect, while newer versions have a Bluetooth 4 (SMART) module and as soon as the app is opened it connects almost immediately.
The app then tracks sleep quality, respiration and heart rate. It shows how many times you got out of bed, when you went to bed and when you got up. Unfortunately currently you need to start the app to do this which means if you forget, say because you're drunk - when you'd really like to know how you slept, it won't measure anything.
If you do remember to pen the app, then you get a nice display of min/max heart rate and averaged breaths per minute and then a visual graph of what kind of sleep you were in.
The next version of the app should automatically connect when near the Beddit sensor and then know when you go to bed and automatically track your sleep. It may also have a live display showing breathing and/or heart rate.
If you have problems sleeping, this won't cure them, but will show you your sleep patterns and maybe help you improve them.
If you're a Misfit user, Beddit also works with the Misfit app (so will use the Beddit to track your sleep rather than the Misfit Shine/etc).
Prices vary widely on-line, but it can be had for as little as £61.