UL stands for "Underwriters Laboratory". It used to be an Insurance Industry organization, but now it is independent and non-profit. It tests electrical components and equipment for potential hazards. When something is UL-listed, that means that the UL has tested the device, and it meets their requirements for safety - ie: fire or shock hazard. It doesn't necessarily mean that the device actually does what it's supposed to, just that it probably won't kill you.
The UL does not have power of law in the U.S. -- you are permitted to buy and install non-UL-listed devices. However, insurance policies sometimes have clauses in them that will limit their liability in case of a claim made in response to the failure of a non-UL-listed device. Furthermore, in many situations the NEC will require that a wiring component used for a specific purpose is UL-listed for that purpose. Indirectly, this means that certain parts of your wiring must be UL-listed before an inspector will approve it and/or occupancy permits issued.