During the life cycle of a cell, although chromosomes are always present, at certain times they can be better visualized so that a karyotype can be made. When a cell is not actively dividing, the chromatin (material of the chromosomes) is difficult to visualize as distinct chromosomes. By the time the cell has finished this stage of its life cycle called interphase, all of its DNA has been copied. It is ready to undergo the M phase including mitosis (copying of the nucleus) and cytokinesis (division of the cytoplasm).
During the M phase, the chromosomes have become more condensed and coiled around proteins called histones. During prophase, they become visible and later, by metaphase, they are at their most condensed. They can easily be visualized under a microscope. At this point, specimens are placed on slides and a stain is added. Next a photograph is taken of the stained chromosomes. Later, they are measured and put into homologous pairs based on their size, with the last pair containing the sex chromosomes.
From the karyotype, one can see if a normal number of chromosomes is present, whether the embryo is male or female and whether there is the presence of certain specific genetic conditions including Down's Syndrome--an extra chromosome at pair 21.