It was first introduced in an article in 1982 in The Atlantic Monthly by 2 social scientists Wilson and Kelling:
"Consider a building with a few broken windows. If the windows are not repaired, the tendency is for vandals to break a few more windows. Eventually, they may even break into the building, and if it's unoccupied, perhaps become squatters or light fires inside. Or consider a sidewalk. Some litter accumulates. Soon, more litter accumulates. Eventually, people even start leaving bags of trash from take-out restaurants there or breaking into cars."
Wilson and Kelling defined 3 factors of why the state of the urban environment may affect crime:
What the theory implies is that if you see an environment with broken windows and other signals of lack of care (graffiti, littering, buildings falling apart), potential criminals feel safe to engage in criminal activities (because they assume a lack of monitoring) and inhabitants and other visitors are scared and avoid being outside.
Or to use Wilson's words (1997):" Therefore, the objective for preventing street crimes is to prevent the first window from getting broken, or prevent the first graffiti marks, or prevent the first drunkard from a public display. This has led to Neighborhood Watch programs and increased police foot patrols. These measures have not had a significant impact on crime, but they have succeeded in making neighborhood residents feel safer."
This theory has been implemented in several situations, one very famous one being the case of New York, where Kelling was hired as a consultant, and where it has been claimed to be the cause of the radical decrease of crimes in the 1990's (see for example the discussion of the NY case by Gladwell in The Tipping Point and the article by Kelling: http://www.city-journal.org/2009/nytom_ny-crime-decline.html).
While the theory has also been implemented with success in Albuquerque, New Mexico, Lowell, Massachussetts, and in the Netherlands, it has also been heavily debated and criticized.
While it might not be the only explanation of the successes (including New York), it does provide interesting insights.
In case you want to know more, here is a study done by the Standford psychologist Zimbardo to test the Broken Windows theory:
"In 1969, Zimbardo placed one 1959 Oldsmobile auto on a street across from the Bronx campus of New York University (a ghetto area), and one on a street in Palo Alto, California near the Stanford University campus (a rather affluent area). "The license plates of both cars were removed and the hoods opened to provide the necessary releaser signals (Zimbardo, 1969)." Within three days, the car in the Bronx was completely stripped, the result of 23 separate incidents of vandalism. The car in Palo Alto sat unmolested for over a week. Zimbardo and two of his graduate students decided to provide an example by using a sledgehammer to bash the car. They found that after they had taken the first blow, it was extremely difficult to stop. Observers, who were shouting encouragement, finally joined in the vandalism until the car was completely wrecked."