To begin, combine your DNA sample, a DNA polymerase, all four deoxynucleotides (dNTP's) and 2 oligonucleotide primers that are complementary to the 3' ends of the two strands of the target sequence.
Heat the sample to 95 C to separate the DNA strands (1: strand separation)
Cool the sample to 55 C so the primers will hybridize with the DNA strands (2: primer binding)
Heat the sample to 75 C so the DNA polmerase synthesizes new DNA strands by extending the primers (3: primer extension)
Repeat as many as 40 times. Because the primers match to both ends of the target DNA, this sequence is preferentially amplified so that it doubles in concentration with each reaction cycle. Thirty cycles yields 2 to the 30th power or 1,073,741,824 copies.
The bacterial polymerases used must be able to withstand the high temperatures required for strand separation. The bacteria Thermus aquaticus lives in hot springs and is a good source.
If the primer used to amplify the target DNA contains an error, all the resulting PCR products will contain the same error. A deliberate use of a mismatched primer is used in site-directed mutagenesis, making a mutated protein. Also the metabolic effect of "knocking out" a gene can give you a clue as to the function of that gene. You don't get any results at all if your primer doesn't anneal.
PCR is used when infection is suspected, to confirm the presence of pathogens. HIV and Lyme disease are routinely diagnosed by identifying unique DNA via PCR.