Interaction Design (IxD) defines the structure and behavior of interactive systems.
Interaction Designers strive to create meaningful relationships between people and the products and services that they use, from computers to mobile devices to appliances and beyond. http://www.ixda.org
User Interface (UI) is a part of the User Experience. It is the design in which the user interacts with the site or application.
It is important that UI be done according to standards of usability "best practices." It is part of the whole User Experience.
User Experience (UX) is how a person feels when interfacing with a system.
UX Designers study and evaluate how users feel about a system, looking at such things as ease of use, perception of the value of the system, utility, efficiency in performing tasks, and so forth.