DNA is contained in chromosomes, which stay in an uncoiled form inside the nucleus, or center, of a cell. In the first stage of the process, interphase, the chromosomes are not yet visible by microscope. DNA replicates, which kicks off the cycle and moves it into prophase. The chromosomes tighten up into a coil and the nuclear membrane dissolves. Microtubules form the spindle apparatus, which are fibers crossing the cell that will later be used to pull chromosomes apart.
Each chromosome contains an area of DNA called the centromere. It's usually on the middle of the chromosome and is a point where two sister (identical) strands of a chromosome meet. On these centromeres, protein plates called kinetochores form. From those, kinetochore microtubules form and move the chromosomes to the center of the cell. In the next phase, metaphase, chromosomes are lined up in the middle of the cell, with the sister strands of each chromosome next to each other.
Anaphase is now set up. The microtubules attached to the kinetochores start to shorten, and this action causes the force that pulls the chromosomes apart. Microtubules shorten with the subtraction of tubulin protein molecules. Motor proteins from the kinetochores move along the spindle, removing the tubulin, which causes the fibers to shorten and pulls the chromosomes apart. Also, microtubules at the poles of the cell move, which also causes the force that separates chromosomes. As the kinetochore microtubules continue to shorten, the chromosomes are pulled to the opposite poles.
Once the chromosomes have reached opposite poles, telophase can begin. This essentially reverses what happened in prophase. The chromosomes uncoil, the nuclear membrane reforms and the spindle disappears, creating two cells. It is clear that the separation of chromosomes in anaphase is the main event of the cell division and the kinetochores drive this process.