In airplane-mode networking is "shut down", so you can use your device e.g. while on a plane.
By default, switching to airplane mode shuts off things like mobile data, cell network as a whole, WiFi, etc., so your device won't have negative effects on sensitive electronics. This is most useful while traveling on a plane (hence its name), as otherwise you might be forbidden to use your device aboard. The same concept is valid for several areas at hospitals, among other places.
While in airplane mode, one can separately activate WiFi if needed.
Another side-effect of airplane mode is that your battery will last much longer, as the networking hardware is no longer using energy to keep your data synced, download advertisements, etc.