Walking through the stages in order
Mitosis splits one nucleus into two identical nuclei in an ordered sequence. In prophase, chromosomes condense and the spindle begins forming. Prometaphase follows as the nuclear envelope breaks down and spindle fibers attach to chromosomes at their kinetochores.
At metaphase the chromosomes line up along the cell's equator. Anaphase pulls the sister chromatids apart toward opposite poles. Telophase reverses prophase: nuclear envelopes reform around each chromosome set and the chromosomes decondense.
Dividing the cytoplasm and the checkpoints that guard it
Cytokinesis usually overlaps telophase and physically splits the cytoplasm, completing the transition from one cell to two. In animal cells a contractile ring pinches the membrane inward.
Checkpoints supervise the whole process. The spindle assembly checkpoint, for example, delays anaphase until every chromosome is properly attached, which guards against mis-distributing chromosomes. Failures in these controls are a recurring theme in cancer biology, where division proceeds despite errors.