Requirements engineering is a term that includes all activities related to requirements - elicitation, analysis, documentation and specification, validation, and management. It might also involve some levels of modeling, ranging from the creation of use case models to more detailed collaboration with system architects and designers.
Requirements analysis is one aspect of requirements engineering that involves reviewing the requirements to ensure that all requirements are useful to building the product. For each requirement, the person or people performing the analysis will ensure they have the characteristics of a good requirement. Some source also consider early prototypes to be part of requirements analysis, since it helps to further understand and clarify requirements given by stakeholders or identify problems with the requirements.
To think of requirements engineering as a staged activity, however, it probably not a good idea. The various activities are ongoing throughout a requirements definition phase of the project. With some regards, requirements engineering also happens throughout the entire project. It just becomes useful to give names to activities to make it easier to discuss with others.