They have little movement monitors that you can clip to their waistband that monitors the rise and fall off their chest. If it doesn't rise for 15 seconds or something like that, you get an alarm. Hopefully it's enough to startle the baby into breathing again, but otherwise at least the parents are alerted as well. There are methods of encouraging the baby to breathe again, but I haven't learned that yet. (The wife and I are expecting, so we're learning all this stuff right now. 16 hour course next weekend... Oof)
Rather than use a clip-on version, go with something like AngelCare's line of under-the-mattress ones. False-positives will be the bane of your existence, so you want to avoid things that can fall off. With ours we usually got alarms when our kids rolled to the very edge of the bed.
We never had any real issues with either of our kids, and only false alarms. But I know my wife got better sleep just because the alarm was there (only in part because I was usually the one to check when the alarm went off).
We stopped using it nearly immediately. It was a glorified audio only monitor for us, the check for movement/breathing was giving lots of false positives and drove us insane.
Other parents even turned on a beep for each breath, made it sound like a hospital. You subconsciously gold your breath if the device beeps with a tiny delay.. beep..beep....???beep
Same one we've used for our three. Awesome product. False alarms only really were an issue the first few weeks of life, as the babies got bigger, it was easier to pick up the breathing patterns.
