You start out with a piece of DNA that has a section whose sequence you're interested in making many copies of, or amplifying. The PCR does this amplification for you.
The DNA (usually double-stranded with two antiparallel strands) is put into a buffered solution containing DNA polymerase (an enzyme that replicates DNA by building new strands that run antiparallel to the template strand), deoxynucleotide triphosphates (or dNTPs, the building blocks for new DNA strands), and oligonucleotide primers (short, single strands of DNA or RNA that are needed for polymerase to begin its work) that adhere to the beginning of the sequence you are interested in on both the parallel and antiparallel strands.
Note that you need a relatively high concentration of dNTPs so that many strands can be made. You also need a higher concentration of primers than regular DNA so that there's a much higher chance that primers and single template DNA strands will stick together rather than the two original template strands. If the template DNA strands stick together and not the template and primer, then the reaction can't happen. The ideal primer has a high complementarity, or match, for its binding site on the template strand, but it must also not form secondary structures.
The whole mixture is put through a thermal cycler (heating and cooling) to get the regular DNA strands to separate and to get the primers to anneal (or attach) properly to the template DNA strands at the sites of interest. Once the primers have annealed, the reaction can begin.
DNA polymerase rides along the template strands, adding dNTPs one at a time to the 3' of the primer stuck to the strand so that you have two new strands growing antiparallel to the template strands. These new strands will be essentially identical in sequence to each template strand's complement. However, they will also be shorter because the primer is not usually designed to adhere to the beginning of the template strand.
DNA polymerase continues to copy until the polymerase reaches the end of the template DNA strand and disengages. What you now have is your original DNA separated in solution as well as shorter copies of the DNA. A second round of replication using the shorter strands produces even shorter strands limited in length to the region between the two primer annealing sites.
The original and new strands can both be used as templates for further copying by DNA polymerase using the same set of primers as before. This means that each round of copying results in exponential growth in the number of DNA strands one has with each passing round of replication.
Unfortunately, this whole process is very difficult to describe in words. The best way to understand it is to see a video of it.