UI Design (User Interface Design ) is a specialized stream of Graphic Design, where the Graphic Designer typically focuses on the Design of User’s Interface and Interactions. This may in certain cases require some specialized knowledge of some extra software/ languages. It is more precise and researched than a typical graphic design solution.
UX Design (User Experience Design ) is a very different animal. It is the domain of researcher, and can be compared to system design/ solution design/ product design, and is actually the name of the process for doing these. It is a process that takes you through bringing up an solution and usually ends at Coding (with UI as part of IT).
UX is a process that has been defined differently by different people. I have tried to come up with an easy to remember method (using mnemonics) and am sharing the same with you. I hope that it is helpful.