On a steam locomotive, a device to enable otherwise fixed driving wheels to have limited motion from side to side. This motion is normally very restricted, on the order of an inch or so in either direction, and generally has a centering force applied.
Without lateral motion devices, all the driving wheels on the locomotive are fixed in place (assuming this is not an articulated locomotive, but this still applies to the seperate units of an articulated). This being the case, upon entering a curve, the entire turning force (ignoring for now any self-centering leading truck which may share in this work) is applied to the flanges of the leading pair of drivers. This force may be sufficient to make the flange hop over the rail, causing a derailment. A similar thing may happen with the last pair of drivers, also.
With a lateral motion device in place, the first pair of driving wheels will take a large portion of the lateral force, but they will be allowed to deflect, allowing the second pair of drivers to also take some of the lateral force. Since the force is shared, the locomotive won't derail; wear on the track and the wheels is also reduced.
The problems with lateral motion devices are mostly that they introduce a lot of complexity into what was otherwise a simple system. By allowing lateral motion of the drivers, lateral motion of the side rods is also required, generally necessitating spherical bearings and a lot more complexity there.
The most common form of lateral motion device was invented and patented by ALCO.
Several extreme forms of lateral motion device exist for locomotives that negotiate very tight curves. These include the Klein-Lindner system, which uses two concentric axles, the inner being driven by the side rods and the outer tubular axle, keyed to the inner, being allowed to rock and slide on the inner, but not turn relative to it. There are also methods whereby the drive to the outer axles is via central gearing rather than outside rods.