PCR is the acronym for the biological testing technique known as the Polymerase Chain Reaction. The process became popular as a DNA study tool because one can amplify (by replicating it) DNA millions of times. If a technician uses PCR to replicate DNA millions of times, that DNA can be used for a variety of purposes. For example, if you add something called «restriction enzymes» and put the DNA in a gel electrophoresis, you get that funny little dot pattern that they use for paternity tests and CSI work.
All organisms use nucleic acids (NA) as the «blueprint» for their genetics. When a cell is dividing and the NAs in that cell are replicating, you start with the double-stranded nucleic acids that must separate during the replication. Now you have two strands of single-stranded NA, in an overly-simplified explanation. Unlinked NAs that compliment the original strands will then anneal (or bind) to the original strands. Polymerase is the biochemical that zips all the unlinked nucleic acids together to make another strand and what you end up with is two double-stranded NA chains.
Another common nucleic acid is RNA, but it's not our genetic material. Ours is DNA. The only differences are: DNA is missing an oxygen on the saccharide backbone (geek talk), and DNA uses Thymine where RNA uses Uracil. Turns out, that's a big difference. DNA is more stable and is therefore the genetic material used in most organisms.
So what's the big deal with polymerase in regards to PCR? Polymerase is used for nucleic acid replication. If you want to amplify human DNA, you have to break apart the two strands so unlinked NAs can fit in there and you need polymerase to link them all together. The problem is that to break apart the two strands in a test tube, you need to heat the sample up to a temperature that destroys polymerase. Shit! Turns out, some genius was studying micro-organisms in the hot pots of Yellowstone and found that those organisms use a form of polymerase that is functional at high temperatures. That guy got a Nobel Prize and rightfully so.
How does it work? You inject a sample of DNA into your test tube and add unlinked nucleic acids and high-temperature polymerase to the sample. In a process of repeated heating and cooling one can copy the strands. Each heating and cooling cycle has the ability to double the strands of nucleic acids.