To run PCR, you need DNA, primers that match the sequence you're trying to replicate and polymerase.
PCR consists of 3 steps that are cycled over and over again:
Denaturation (~94C) At this step, DNA 'breaks apart', splitting from a double helix into single strands
Annealing (~60C) Primers bond to the single-stranded DNA
Extension (~72C) Polymerase compliments the DNA, synthesizing strands that are of the target sequence
Each of these phases can be 20-30 seconds long and repeated 30+ times, depending on the protocol. Most protocols also suggest having a longer initial denaturation step and a longer final extension step.
A simple tutorial:
There's also a bunch of related resources here: http://www.lab-manual.com/lm_209.htm
The results of PCR can be visualized using gel electrophoresis. DNA samples are loaded into a gel, and a high voltage is applied across it. Because DNA is negatively charged, it will travel through the gel at different speeds depending on its size. This process will effectively separate out the pieces you want, and you can see them by staining the gel. Here's a good tutorial
and if you're trying to DIY it, the Macgyver Project
is a pretty good resource.
PCR can be performed using 3 water baths (each kept at one of the three temperature settings). A human could physically move the samples from one bath to the next 30+ times. PCR machines were developed to automate the process, but most lab-quality ones cost thousands of dollars. But they don't need to! Today there is a growing number of open source PCR projects, among them OpenPCR
($200), and the Coffee Cup PCR