A trisomy is caused – in most cases, and for Down Syndrome 95% of cases – by non-disjunction.
Non-disjunction occurs during meiosis (meiosis being the process by which sperm and egg cells are formed – each with typically 23 chromosomes).
Typically, in meiosis, the parent cell is meant to split, and give one of each chromosome to each reproductive cell, when non-disjunction occurs, both copies of a chromosome end up in the same reproductive cell, which then combines with the single chromosome of the other parents cell during fertilization, leading to a trisomy.
This is a very basic overview, and genetics and biology both go far more in depth as to the whys and hows of cells, and what they do. This is the layman’s version of what causes a trisomy.