In this process, a strand of DNA is added to a solution containing: individual nucleotides, DNA polymerase, and synthesized sequences of nucleotides called primers which define what section of the DNA is amplified. The solution is then heated to nearly 100 degrees Centigrade, at which point the hydrogen bonds between the two DNA strands break.
While cooling, the synthesized primers bind to their complimentary regions on the separated DNA strands. From there the DNA polymerase recognizes the primers as starting points and begin moving down the DNA strands, adding a complimentary nucleotide for each nucleotide on the template DNA. The end result is two new DNA molecules, each one single stranded before the primer and double stranded after the primer.
When the solution is heated again, the new DNA strands melt, allowing primers to attach to the appropriate site, and the process continues until one decides enough DNA has been made. This shouldn't be very much time, since the amount of DNA is doubled each time the cycle is enacted.
One problem posed by this method is that most DNA polymerases dissociate, or unfold, well before 100 C. To solve this, DNA polymerases from a bacterium living in hot springs, which have evolved to function in boiling water, is used.
Hope that helps.