A Post-industrial economy is one in which technology has advanced to the point that the economy is no longer characterized by the need for large numbers of factory workers. In other words, productive capacity and efficiency have increased to the point of mass lay-offs.
This is similar to the process which occurs as an economy moves from an agricultural base to an industrial one–when huge portions of the population will leave the countryside for new work in the cities. Some descriptions say a post-industrial economy means a shift to the service sector and information based jobs. While this is accurate to a degree, it does not capture the level of dislocation–mass factory work and the mass consumption it allowed for are the two halves of the self-reenforcing cycle that has defined modern economic life. The comparison to the dislocation of the rural population at the beginning of industrialization is better.
One issue that does not seem to be well described and dealt-with, at least in much of the public policy world, is what we might call the distribution problem–as technology advances, the same or a larger aggregate amount of industrial products are being produced, but the simple system by which the aggregate mass of workers who produced the products made enough to buy them no longer works. Karl Marx accurately described this as the source of the recurrent crises in capitalism, although in recent years it seems they might be described as a state of chronic malaise. Some economists have argued that this is a source structural unemployment–unemployment that will not go away as an upturn in the business cycle increases aggregate demand.
But whatever it's nature it is clearly a new stage in economic development.
See also Towards A New Dialectic.