How does the US National Science Foundation (NSF) research funding system work?
I'm most familiar with the NSF mathematical sciences research postdoctoral fellowship, which I was lucky enough to get 15 years ago.  The application process is described in detail in NSF's formal solicitation, but here's an executive summary:


Unlike most other NSF funding, this program is limited to US citizens and permanent residents.  (NSF's regular research grants formally fund the institution, not the PI, so non-American PIs can win grants; on the other hand, NSF's graduate research fellowship has a similar citizenship requirement.)  There was a parallel program for international postdocs for several years, but it seems to have been retired.
Applicants must be within two years of their PhD and must have no previous US federal grant funding.  In practice, this means each applicant can apply at most three times: once just before graduating and twice after.  Applicants do not need an academic affiliation when they apply.  (Again, the fellowship funds the applicant, not the institution.)
The application itself requires a bunch of NSF boilerplate, but the main content is an abbreviated (3-5 page) research proposal, a one-page project summary, and four recommendation letters (submitted separately by their authors).  As usual for NSF, the proposal absolutely must contain a paragraph explicitly labeled Intellectual Merit and another paragraph explicitly labeled Broader Impact.
Applicants also need an agreement from a sponsoring scientist/mentor/advisor.  The sponsor separately submits a statement describing their proposed mentorship role, research opportunities available in the hosting department, and promised infrastructure (office, computer support, libraries, specialized equipment, etc.).  The sponsor cannot also write a recommendation letter.
All applications are submitted electronically through NSF's FastLane web site.  Considering how much FastLane has to handle (basically everything NSF does), it works amazingly well.
The applications are reviewed by a panel of mathematical scientists (or more likely by multiple panels, given the number of applications).  Panelists are invited by the program director, but in my experience, NSF program directors are happy to hear from volunteers! 
Each panel recommends and ranks a subset of the applications they review.  Standard NSF practice is to fund all applications strongly recommended by the panel, and some applications with weaker recommendations, depending on available funds and other criteria.  (NSF is deliberately vague about these other criteria, but gender, ethnic, and geographic diversity are likely guesses.) 
Applications for fellowships to begin in Fall 2013 are due October 19, 2012.  Yes, this is really early.  Winners are usually notified in February and announced on the NSF web site soon thereafter.  All applicants receive a summary of their panel reviews.
The total award amount is $150,000, distributed over a two-year period.  Most of that is salary, but about $30,000 is set aside for stuff like equipment, travel, and benefits from the host institution.  Frustratingly, fellows are neither employees of the hosting institution, employees of NSF, nor formally self-employed; the US Treasury Department simply injects $5000 into your bank account every month.  So good luck with taxes!
Fellows need to submit progress reports once per year and a final report when the fellowship ends.