In 1866 John Langdon Down, a superintendent of an asylum for children, noted that some children with mental retardation shared a common set of features, namely flat faces and epicanthal folds. Because of these features, he called this form of mental retardation mongolism* (he contrasted this to cretinism, now known as hypothyroidism) .
In the 1960s this was deemed politically incorrect, and it was changed to Down's syndrome. In the 1970s, an American revision of scientific terms changed it to Down syndrome, and in Europe it's often called simply Down's. In 1959 Jerome Lejeune and Patricia Jacobs (working independently) determined the cause of Down's to be trisomy of the 21st chromosome, leading to the name trisomy 21.
Other characteristics of trisomy 21 are congenital heart defects, a predisposition to leukemia, Alzheimer-like changes in the brain at around 40 years of age, simian crease (a crease in the palm), and abundant neck skin.
There are three conditions resulting in trisomy 21. The first, and most common, is simple nondisjunction. This accounts for about 90% of cases. When a cell undergoes meiosis (to make sperm or ova), chromosomes are supposed to split up, depositing 23 chromosomes in each daughter cell (it's more complicated than that, go read meiosis). Sometimes a chromosome doesn't split, and one gamete receives only 22 chromosomes, the other receiving 24. If the gamete with 24 chromosomes pairs up with a normal gamete to form a new zygote, this zygote has three copies of a chromosome (one from the normal gamete, and two from the irregular one).
Next up, Robertsonian Translocation accounts for 3-4% of cases. In this case, two chromosomes develop breaks, and material is traded between them. The breaks most often occur in the 14th and 21st chromosomes. In this case there are still 23 chromosomes in the gamete, but there is more information from the 21st chromosome than there should be. In some cases, only part of the 21st chromosome is included, and thus it is possible to have partial trisomy 21.
Finally, mosaicism account for all other cases of trisomy 21. This simply means that there are at least two separate cell lines. Some of their cells have trisomy 21, either from nondisjunction or from Robertsonian Translocation, and others don't. This results from a mistake in mitosis, not meiosis, and the original zygote did not have trisomy 21.
The chromosomes determine what part of the DNA gets expressed where/how/when, and having an extra chromosome of one type leads to overexpression of the genes in that chromosome. We're still working to find exactly what genes are on the 21st chromosome.
*after the people of Mongolia.