Getting a UL stamp means you meet a number of safety tests for your equipment. In the US a UL stamp is only 'required' on some electrical equipment in some markets, generally items running on main power but there are some exceptions and only when its being sold or placed in some public areas, for instance a school. Even then a UL stamp is completely optional, however a local inspector may not allow a non-UL device in a building. Needing to be UL listed is completely a political issue not a technical one.
The documents that cover a UL listing are long and cover many issues from shock prevention, reducing the chance of a fire, RF emissions, etc.
As one example you could use a UL listed power supply to power a low voltage device, something like a laptop brick, but if you place that brick within an enclosure it is no longer UL compliant.
UL listing is really a hoop you need to jump through for safety compliance for a product your selling as defined by the market your selling that product to. I wouldn't try to put a non UL listed part in something like a school but for some applications its not needed.
Getting a UL stamp is expensive. The European version of the UL stamp is the CE stamp and it actually goes farther in various directions than a UL stamp does.
I wouldn't bother worrying or caring about a UL listing unless you know that your market or installation cares about that, the best way to determine this is to call you local electrical inspector and ask him about the issue at hand.