Aneuploidy is when a cell has more than the normal number of chromosomes. A human cell with the exception of red blood cells and gametes, has exactly 46 chromosomes. (Recall our strands of DNA are divided into chromosomes and they are subsequently divided into genes.) Sometimes aneuploidy occurs at birth such as with Down's Syndrome. More often it occurs infrequently within an otherwise average healthy human body.
Cancer Cells' Chromosomes Hint at New Treatments explains that researchers have determined that aneuploid cells have different characteristics from normal cells. For example, they divide much slower, need additional energy, and grow larger than normal cells. This observation will allow the development of medical regimens that can kill these weaker cells.
The connection to cancer is described in The aneuploidy theory of cancer and the barriers to its exposure which estimates that 90% of human cancers are aneuploid. As the paper describes, it could be 100%.
Radiation is known to cause aneuploidy. More importantly, carcinogens are known to cause aneuploidy. The prevailing theory has been that gene mutation has been responsible for the development of a tumor. An emerging hypothesis is that cancers are really caused from aneuploidy. The latter argues that it is chromosomal imbalance caused by thousands of genes (from an extra chromosome) that causes a cell to become cancerous rather than a handful of genes in the genetic mutation theory.