The replication of the DNA molecule, is a process which lies at the very heart of life it self. The process is known as mitosis and involves the separation of the DNA in a cell to become two unique instances of the same thing. The whole process is facilitated by enzymes that are produced by RNA from the DNA in a process known as transcription.
If you were to look at DNA in the nucleus of a cell under a microscope, it would look very much like a bowl of spaghetti, all coiled around itself and other DNA molecules, and even knotted up in places. Before the replication of the DNA can begin, it needs to be straightened out, unknotted and uncoiled. An enzyme called "Topoisomerase" takes care of this job and the copying is ready to begin.
Once a DNA double helix strand has been readied by the topoisomerase, another enzyme called "Helicase" begins to separate the metastable hydrogen bonds between nucleotide base pairs that hold the two strands of the DNA molecule together. The helecase enzyme basically unzips the DNA double helix, producing two mono helix strands of the DNA. Now a third enzyme called "Polymerase" on each of the now separated strands, rebuilds the other half of the DNA molecule by complimentary base paring. The amino acids that are used to rebuild the base pairs, called "primers," come from the cytoplasm soup in the cell nucleus. The polymerase essentially grabs the appropriate base pair and then joins it in the nucleotide by hydrogen bond. It so happens that sometimes Polymerase makes a mistake and joins a Uracil primer instead of a Thymine one. When this occurs, some other enzymes correct what are essentially DNA spelling errors. On rare occurrences however, the error is not corrected, changing the protein that a sequence of nucleotides code for, and the cells function may be impaired or compromised. Cancers are conditions which stem from such errors during mitosis.
While the polymerase is during its thing, another enzyme called legase joins the new nucleotide segments to the DNA sugar phosphate backbone. It takes quite a while for an entire DNA molecule to be copied and when it is done, assuming all goes well, you end up with two identical copies of the original DNA molecule. Each of the copies contains half of the original the other half being newly synthesized.
The process is completed when an enzyme called "telomerase" glues together the telomer nucleotides at the end of each DNA molecule, making sure that it won't unzip on its own. So that in a nut shell, is how a DNA molecule replicates it self.