Physics describes evolution: it tells how, given some initial state, a system will look at later times. To describe the earliest moments of the Universe in quantum cosmology, we need to have some idea about the initial conditions, or "boundary conditions" at the beginning of time. One proposal, due to Hartle and Hawking, is that "the initial boundary condition of the Universe is that it had no boundary."
Popular books describe this proposal with varying degrees of accuracy. (See PBS's " Stephen Hawking's Universe" for a fairly good example.) By forgetting that the "no boundary proposal" is a quantum mechanical description, though, these popularizations can sometimes be misleading.
In particular, it's worth remembering that a quantum mechanical object does not have a unique, well-defined "history." For a particle, for instance, such a history would be a trajectory -- position as a function of time -- and would determine both the particle's position and its momentum at all times. But by the Heisenberg uncertainty relations, this cannot be done: we can never simultaneously exactly specify a particle's position and momentum.
The Hartle-Hawking "no boundary" proposal is based on the path integral, or "sum over histories," approach to quantum mechanics, in which a probability amplitude is computed by taking a weighted sum over all possible histories that lead from an initial condition (in this case, "nothing") to a final state. In a certain approximation, this sum is dominated by a "history" in which the Universe initially has a positive-definite metric -- thus the frequent references to "imaginary time." But neither this nor any other single history represents "the way the Universe really evolved."
- Steve Carlip (Physics, UC Davis)