Without cell division, there would be no life on Earth. Every species creates daughter cells from a mother cell. The most common method of dividing is a process called mitosis. Mitosis duplicates and splits the DNA -- the chromosomes -- within a cell so each daughter gets a full set. To finish the job, there's a final step called cytokinesis in which the cytoplasm is divided between the daughter cells.
Although mitosis in different types of organisms shares many characteristics, there are some differences. In animals, each cell combines tiny fibers to create a cord called the spindle. The spindle stretches through the middle of the cell, DNA duplicates and condenses into visible chromosomes, then the nuclear membrane breaks down. Fibers from the spindle connect to the pairs of chromosomes and pull them to different sides of the cell. That completes the preparation for division.
The Cleavage Furrow
After the division of the nuclear material is completed, the cell develops a ring around its center. The ring is a narrow furrow called the cleavage furrow. The direction of the spindle directs the orientation of the cleavage furrow. If you think of the spindle as a long pole, the cleavage furrow is a ring around the center of the pole. This is important, because the cell is like a ball, which means it could split in half along any line through the center. But if the cleavage furrow were oriented along a different axis, the chromosomes might not divide equally between the different halves.
The Contractile Ring
Imagine you have a round balloon. If you wanted to shape it into separate halves, you'd place your hands around the middle of the balloon and squeeze. Cytokinesis happens the same way, except that there are no hands from the outside to give the cell a squeeze. Instead, an internal ring built of the proteins actin and myosin pulls tight. Actin and myosin are the same proteins that contract your muscles, and the ring they make around the center of the cell is called the contractile ring. The contractile ring shrinks right through the center of the cell, evenly dividing the cytoplasm between the two halves.
Cell division for egg cells proceeds in a different fashion. The overall process of cell division for sex cells is called meiosis. Meiosis has several differences from mitosis -- the biggest being that one mother cell with a full complement of DNA ends up as four daughter cells, each with one half the full complement of DNA. With egg cells there's one more big difference. Cytokinesis happens asymmetrically. That is, rather than a cell division producing two same-sized daughter cells that equally share the cytoplasm, each division provides the vast majority of the cytoplasm to what ends up as the egg cell. The other entities are called polar bodies, and they don't contain enough cytoplasm to survive.