An excellent introduction is The Rational Unified Process Made Easy by Per Kroll and Philippe Kruchten.
Another very readable introduction to the RUP is The Rational Unified Process An Introduction by Philippe Kruchten, its lead architect.
Applying UML and Patterns: An Introduction to OOA/D and the Unified Process [Larman01] takes a case-study approach to introducing the UP and OOA/D. The concepts and workproducts are introduced incrementally over a series of case-study iterations.
As a caution, there are a number of so-called UP and RUP texts that give incorrect descriptions.
Worthwhile readings that influenced the UP creators:
"A Spiral Model of Software Development and Enhancement" by Barry Boehm, ACM SIGSOFT Software Engineering Notes, August 1986 (the reprint in IEEE Computer, May 1988 is often cited). A widely cited paper that describes the spiral model of iterative and risk-driven development.
Tom Gilb's iterative Evo method, described in Principles of Software Engineering Management, 1988.
Grady Booch's Object Solutions Managing the OO Project, 1995.
"Anchoring the Software Process" by Barry Boehm, IEEE Software, July 1996. Describes the milestones from which the inception, elaboration, and construction phases were inspired.
The PMBOK (www.pmi.org). Some of the Project Management discipline was influenced by the PMBOK.