The Fair Copyright in Research Works Act is supposedly intended to protect the intellectual property of scientists and their funding sources. Right now the NIH requires that all research funded by them be made available to the public for free after a certain proprietary period (in addition to their usual publication in peer-reviewed journals). Passing the bill would allow the NIH (and other federal groups such as the NSF, etc.) to remove this requirement from federally funded researchers, so that the researchers could choose to (or additional private funding sources could require them to) only publish their work in expensive peer-reviewed journals.
There is some debate about whether this is really as bad as it sounds. On the one hand, it sounds like the Act would allow the stifling of scientific communications and would mean that the American public would have to pay twice to see the work that their taxes paid for. On the other hand, it's my understanding that all federally funded research is required to be public domain and it's not clear that this Act would counteract that, or that the NIH and other federal groups would choose to do what the Act would allow. In addition, it might allow researchers to use multiple funding sources for a project which they may be unable to do right now - pharmaceutical companies may require NDAs for their work which the NIH regulations currently rule out since the researchers have to publish publicly.
Some links for more reading
The Act itself, Library of Congress
Phil Plait/Bad Astronomer's first post
His second post
A lawyer's opinion
Financial Times article