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DNA stands for deoxyribonucleic acid. DNA is the building block on which almost every form of life on Earth is based. DNA is essentially the blueprint for living organisms.
DNA is an organic nucleic acid that exists in all cells. It is composed of three basic components wrapped into a double helix structure: a phosphate group, a five-carbon sugar, and nitrogen bases. The phosphate and sugar (in this case, deoxyribose), act as the walls of the helix, while genetic information is stored in the sequence of nitrogen bases that connect the two sides of the helix. There are four basic nitrogen bases in DNA: adenine, cytosine, guanine, and thymine, often abbreviated to A, C, G, and T. Adenine can only bind with thymine, and guanine can only bind with cytosine. Massive strands of nitrogen bases can carry vast amounts of information, which is primarily used for the construction of proteins.
A considerable proportion of an organism's DNA is so-called "junk" DNA, as it does not appear to have any specific purpose. As the exact way in which DNA is interpreted is not fully understood yet, it is quite possible that some junk DNA performs a vital role.
 Human DNA
The fact that other than identical twins no two people have the same DNA allows determination of identity. So-called "DNA tests", however, do not involve a comparison of all of the DNA; a complete sequencing of the genome is rather expensive. Instead, regions of high variability in the genome are identified, and those areas are compared. The number of regions varies depending on how thorough the test is. With enough regions tested, the chances of a false positive can be made extremely small. Sometimes, this probability is given as less than one in a billion, but the exact probability cannot be calculated without assuming that the regions vary independently, which is not necessarily the case. Furthermore, it's a bit silly to talk about the chances of an error being one in a billion, when there are all sorts of ways that there could be a false positive, such accidental contamination, deliberate planting of evidence, et cetera. So-called "identical" twins are often assumed to have identical genomes, but research is still ongoing as to what extent that is the case. In fact, there can even be different DNA within the same person.
Similarly, DNA can be used to determine ancestry, especially in cases where paternity has to be ascertained.
Mitochondrial DNA (which is only passed down the female line as mitochondria in the sperm are lost with the flagellum or selectively destroyed after fertilisation) is used to study human evolution because it changes more slowly than cellular DNA. It has been used to trace the origin of female humans back to prehistoric Africa. Mitochondrial DNA is also used to trace the migration of humans after they left Africa. However, concern has arisen about mitochrondrial DNA dispersing through populations at a rate different from nuclear DNA.
 How does DNA actually, er.. work?
Let's not get too complicated here, or too hidebound by detailed facts. Here's the telegram version.
DNA replicates by unwinding into two strands. Each strand then picks up all its complementary A,C,G, or T bases from stuff that's lying around. The result is two copies. NEVER MIND about mutations. NEVER MIND how it picks up the bases.
DNA synthesises proteins—that's what we're mostly made of—by a different kind of unwinding. In this type, the complementary strand is the same sequence, but a molecule of mRNA. NEVER MIND about Uracil.
mRNA then chuffs off and makes the protein. Every triplet of RNA bases encodes a specific amino acid. The amino acids are strung together in a long chain and there's yer protein. NEVER MIND about tRNA. NEVER MIND about start/stop codons. NEVER MIND about secondary and tertiary structure.
Don't even think about all the other things regulatory sections of DNA produce like lncRNA's, snRNA's, snoRNA's, miRNA, rRNA, and a slew of other regulatory and structural RNA's.