jcp - Failsafe pretty much means designing things to fail in a safe manner as opposed to them failing in a dangerous manner.
For example, say you have a sensor at the end of a pallet conveyor. When it detects a pallet at the end of the conveyor it stops the conveyor until the next system removes the pallet. If the conveyor were to continue running, the pallet could topple over, hurt people, and damage product or equipment. For this example we'll say that the sensor is a retro-reflective photo-electric sensor.
In this scenario, we want to design the system so that if the sensor fails for whatever reason (cable gets cut, sensor gets knocked out of line, reflector gets broken, etc.) that the conveyor will stop.
An easy method for this is to use a normally closed output on the sensor. So when a pallet is detected the circuit opens. This also means that if the cable gets cut or the sensor gets knocked out of line, the circuit will open. If the sensor circuit opens for whatever reason, we will stop the conveyor.
This way, we have designed this system to fail in a safe manner. If anything goes wrong with this sensor, the conveyor will go into it's stopped state.