“UI” stands for “User interface”. Traditionally, that means the actual buttons, text, and pixels that appear on the screen. So the UI designer would be responsible for everything about how a web site or application appears in the interface – this would include visuals (e.g. “Are the buttons pink or blue?” “Do they have gradients or are they flat?”) as well as overall architecture of the page (e.g. “We should put the Sign Up button at the top of the screen because it’s the most important thing”).
“UX” stands for “User experience”. This is a much broader and higher-level discipline than UI, because it includes not just the interface, but also all the systems and interactions that support it. For example, the UX designer should care about what happens when a frustrated user calls the help desk as well as how pixels appear on the screen.
Interestingly enough, because this role can be so high-level, sometimes the UX designer usually won’t pay so much attention to pixels. Instead, he or she will work with either a visual designer or a dedicated UI designer (maybe a junior person on his/her team?) to polish the visual execution of the user experience.