What's a dangling modifier?

Here's an example taken from the pages of the New York Times:

"Now, after starting a new life in Altamira, the government is telling her she needs to leave again, this time to make way for the Belo Monte dam, which will flood a large swath of this city, displacing thousands of people." (August 15, 2010)

"A classic dangler. She, not the government, started a new life."

The example and the comment are taken from the Philip B.Corbett's After Deadline blog, which is published on Times' website every Tuesday.  After Deadline examines questions of grammar, usage and style encountered by writers and editors of The Times. It is adapted from a weekly newsroom critique overseen by Philip B. Corbett, the associate managing editor for standards, who is also in charge of The Times’s style manual.