Down's syndrome is most often caused by having a full or partial ('mosaic') third copy of chromosome 21 when you should just have 2 copies. This is why Down's syndrome is sometimes referred to as 'trisomy 21' (trisomy means 'third copy'). This extra copy means certain genes are over-expressed, which is why people with Down's syndrome will often share physical, cognitive and behavioural features.
A mother's age has a bearing on the possibility of the syndrome in their child - the risk of having a child with Down's syndrome increases markedly the older the mother is.