NSF proposals: What exactly is wanted in the Intellectual Merit and Broader Impact sections?
I suggest the first port of call for an explanation is with the NSF itself. For Broader Impacts, I would start with this Dear Colleague letter on the Broader Impacts Review Criterion.


  The NSF Grant Proposal Guide uses a series of questions to illustrate
  the Broader Impacts criterion: “How well does the activity advance
  discovery and understanding while promoting teaching, training, and
  learning? How well does the proposed activity broaden the
  participation of underrepresented groups (e.g., gender, ethnicity,
  disability, geographic, etc.)? To what extent will it enhance the
  infrastructure for research and education, such as facilities,
  instrumentation, networks, and partnerships? Will the results be
  disseminated broadly to enhance scientific and technological
  understanding? What may be the benefits of the proposed activity to
  society? “
  
  These questions help to assess the potential of the proposed activity
  - beyond the research, per se - to benefit the Nation. Thus, the Broader Impacts criterion speaks directly to the mission of the
  National Science Foundation, “To promote the progress of science; to
  advance the national health, prosperity, and welfare; and to secure
  the national defense.” (NSF Act of 1950).


For even more recent descriptions, review Chapter II - Proposal Preparation Instructions of the NSF Proposal &amp; Award Policies &amp; Procedures Guide, Which includes these parts:

Intellectual Merit


  The Project Description must contain, as a separate section within the
  narrative, a section labeled "Intellectual Merit". The Project
  Description should provide a clear statement of the work to be
  undertaken and must include the objectives for the period of the
  proposed work and expected significance; the relationship of this work
  to the present state of knowledge in the field, as well as to work in
  progress by the PI under other support.
  
  The Project Description should outline the general plan of work,
  including the broad design of activities to be undertaken, and, where
  appropriate, provide a clear description of experimental methods and
  procedures. Proposers should address what they want to do, why they
  want to do it, how they plan to do it, how they will know if they
  succeed, and what benefits could accrue if the project is successful.
  The project activities may be based on previously established and/or
  innovative methods and approaches, but in either case must be well
  justified. These issues apply to both the technical aspects of the
  proposal and the way in which the project may make broader
  contributions.


Broader Impact


  The Project Description also must contain, as a separate section
  within the narrative, a section labeled "Broader Impacts". This
  section should provide a discussion of the broader impacts of the
  proposed activities. Broader impacts may be accomplished through the
  research itself, through the activities that are directly related to
  specific research projects, or through activities that are supported
  by, but are complementary to the project. NSF values the advancement
  of scientific knowledge and activities that contribute to the
  achievement of societally relevant outcomes. Such outcomes include,
  but are not limited to: full participation of women, persons with
  disabilities, and underrepresented minorities in science, technology,
  engineering, and mathematics (STEM); improved STEM education and
  educator development at any level; increased public scientific
  literacy and public engagement with science and technology; improved
  well-being of individuals in society; development of a diverse,
  globally competitive STEM workforce; increased partnerships between
  academia, industry, and others; improved national security; increased
  economic competitiveness of the U.S.; and enhanced infrastructure for
  research and education.


The Short Version

I think about it like this: intellectual merit is the "contribution" part of a potential publication or set of publications; broader impact is what administrators would like you to talk with the media team about so they can make a press release about how their people/funding are/is making the world a better place. In other words, the merit is the work itself and its findings, and the impact is what the effect would be on some defined portion of society if you succeeded in doing what you propose.

Both parts should be about what you are specifically proposing to do now (the thing the proposal will fund, and preferably with early results to support some claims), and should be deeply and believably connected to the research you are proposing. The farther out and more vaguely connected it is, the less compelling and believable it is likely to sound. "These findings could be built on to eventually obtain world peace in the next millennia" is not very compelling, for example.

Grandiose claims of widespread benefits must trade-off against believability and practicality. The more people you could help, in science and outside in general society, the better - but you should make the link very explicit and clear.

My source for the above is being involved with a few proposals in preparation, extensive reading of NSF program materials, preparing and succeeding with a fellowship proposal, and being involved with a few grant processes that were modeled off NSF criteria in the general area of technology/engineering/psychology. If you see any disagreement between my opinion and official NSF program materials, believe the NSF materials over me.