It assumes that the new ULs need to be created dynamically
web designing

Yes, jQuery itself is a library for extending JavaScript to manipulate DOM objects and do other common JS things like AJAX.

JQuery-UI is actually a plugin to jQuery that adds some more animation effects and popular UI widgets like dialogs, datetime picker, sliders, etc...  

There are tons of plugins available for jQuery: plugins.jquery.com

Ghmm. I think it is quite simple. JQuery alone is JS framework. And jQuery UI is extended jQuery framework (or we can say - framework of the framework :D )