Are there any differences between the Rational Unified Process (RUP) and the Unified Process (UP)? What does "Rational" mean, in this context?
The Unified Process is a generic name for a family of process models that meet a number of criteria, such as being iterative and incremental, driven by use cases, and focus on addressing risks early. It defines four project phases: Inception, Elaboration, Construction, and Transition.
The Rational Unified Process is a refinement of the Unified Process that was created by Rational Software (now owned by IBM). It uses a series of software tools along with a process framework to define how to carry out the activities needed to run a software project, yet still provides a framework for tailoring to meet an organization's (or team's) needs.