What happens in Down Syndrome (trisomy 21) is what is referred to as a "mieotic nondisjunction", which is a failure of the sister chromatids to separate when they are supposed to separate. This would produce a sex cell, either sperm or egg, with 24 chromatids instead of the normal number, which is 23. The chromatid that is the extra chromatid is the 21st chromatid, which is why Down Syndrome is referred to as trisomy 21.
In interphase I, each chromosome makes an identical copy of itself. In anaphase I, the chromosomes separate from their homologous partners and migrate along spindle fibers to opposite ends of the cell. In telophase I, the nuclear membrane reforms and the cell divides. So far, so good. It is in the second stage of meiosis the error occurs. The chromosomes are not copied again.