Down syndrome is simply an extra Chromosome on the 21st pair. There are 3 types of Down syndrome:
Nondisjunction - This is the most common form. When the cell first splits, the 21st chromosome (from either the father or mother, but typically mother) is duplicated. Every cell has 3 of the 21st chromosome and 47 chromosomes all together. This form is a random occurrence.
Translocation - This is caused when a piece or all of the 3rd chromosome attaches to another chromosome instead of existing on it's own. Technically, the person has 46 chromosomes, but the extra information is still present. This is very rare, but can be inherited. In this case, the parents will be tested.
Mosaic - In this case, only some of the cells have 47 chromosomes. So, at some point, there was a split where the 21st chromosome was duplicated, but the duplication did not happen with the first split, as in nondisjunction.