Real-time PCR is a technology advancement based on polymerase chain reaction (PCR).  To understand it, we need to understand the mechanism of PCR.
PCR is an in vitro method to enzymatically synthesize a specific DNA sequence with a pair of oligonucleotide primers that hybridize to the opposite strands of the DNA and flank the target sequence.   For each PCR cycler, there are typically three steps:

1. Denature: With a high temperature, usually 95°C, the double stranded DNA is separated into single stranded DNA.
2. Annealing: The primers will bind to the complementary sequence, which forms the start point for DNA polymerase to extend DNA.
3. Extension: DNA polymerase extends a complementary sequence from the primer along a strand of DNA.

After each PCR cycle, the targeted DNA sequence, also called template, is replicated.  Initially, the PCR is accomplished by using Klenow fragment of E. coli DNA polymerase I to extend the annealed primers.  With the discovery of thermal stable Taq DNA polymerase isolated from Thermus aquaticus and the introduction of thermal cycling devices, the PCR cycles can be automatically processed.  This methodology significantly advances our understanding of genomics, and thus was awarded the Nobel Prize in 1993.  The following animation will help you to visualize and understand the mechanism of PCR.

To verify that a correct DNA sequence is amplified, the PCR end product usually will run through an argarose gel electrophoresis.  In a gel, ethidium bromide (EB) is used to bind double stranded DNA; it emits an orange fluorescent light when it is observed under an UV light.  Along with DNA size standard ladder, the size of DNA fragments from PCR can be visualized.  A typical gel image is shown here.