The birth of white flight
came in the 1950s and 1960s, after school desegregation
brought on by the 1954 Brown v. Board of Education
Before the desegregation orders, many upper- and middle-class whites lived in certain areas of American cities -- and here I'm not just talking about metropolitan areas, I'm talking about the actual municipalities. Schools were segregated both de jure and de facto, as white schools would be in white areas, and black schools would be in black areas.
Suburbs were already a going concern, as the men coming back from World War II had often married on arrival and settled into innumerable Levittowns in counties outside the city limits. When orders came to bus students to schools outside their home neighborhoods in order to achieve certain racial percentages of enrollment across the board -- particularly when white students were being bussed to formerly black schools (which were underfunded and ill-kept) -- well-off white parents saw no way short of moving to keep their kids from this situation.
So they did move, out to the suburbs where the county's racial mix was so overwhelmingly white that very little bussing had to take place. This was the first wave of white flight; meanwhile, as mentioned in SheThing's WU, property values in the areas they vacated dropped, allowing more economic lower classes (usually minorities) to move in.
It is difficult to understate the effects of white flight on the development of U.S. cities. With people moving out, the retail industry was quick to follow suit; this was where the downfall of downtown shopping areas began. Next to go were the offices; if the educated employees are all living in Annandale, it's a lot shorter commute to Tysons Corner than to Constitution Avenue (or so one would think), so the employers started going there as well. Thus, the ring city movement began -- an oft-cited anecdote claimed that in the late 1990s, Tysons Corner, Virginia had more office space in square feet than downtown Philadelphia, PA. The initial rings have begun to expand now as inner suburbs take on the same demographic mix as the nearby city limits. Donut-shaped metro areas remain, where the only need to venture into the city center is a convention, sporting event or tourist attraction.
In the late 1990s and early 2000s, urban renewal efforts have revitalized small sections of downtown areas: a new corporate office here (with an underground, guarded parking garage to keep the riffraff away from the Volvos and BMWs), a condominium high-rise or apartment building there, a stadium development several blocks over. There is some chance that this may reverse the effects of white flight, but don't bet on it -- most of those young professionals living in the city are doing so for the reverse commute benefit or (for the singles) the nightlife, and will get out of there as soon as they are married, have kids, and realize the sad shape of the local schools. Then the city is back to square one.