Cytokinesis is the process of dividing a cell into two new cells. It usually follows immediately after mitosis, which is the process of replication of the nucleus. Since the nucleus contains the DNA, which is critical to the functioning of the cell, replication of the nucleus is a well organized and precise process. The dividing up of the rest of the cell's contents is much less precise.
In animal cells, a ring made of myosin II and actin fibers forms around the equator of a cell just as telophase (the last step in mitosis) is occurring. The myosin II creeps along the actin, drawing the ring tighter and tighter and forming a cleavage furrow around the cell's mid line. Septin filaments form an abscission zone which causes the daughter cells to part completely from one another.
Plants cannot form a cleavage furrow because they have rigid cell walls. Cytokinesis in a plant cell results from the formation of a cell plate which develops between the daughter cells, expanding outward until it contacts and joins the cell walls. The cell plate begins as a remnant of the mitotic spindle, and it is then built up with cellulose, pectin, and other cell wall components.