In prokaryotic cells, also known as bacteria, the reproductive process is known as binary fission. There are no organelles to reproduce except the ribosomes, which make protein, and the DNA is one circular loop that is distributed evenly throughout the cell's protoplasm. The cell splits evenly into two new daughter prokaryotes.
In eukaryotic cells, or cells that do have a nucleus and organelles, there are two separate stages of the cell cycle, three if you count cytokinesis at the end when the cell actually splits into two daughter cells. Interphase is the first part, which is termed as the cells normal life existence, where it carries out functions necessary for life. Mitosis is the other stage, which is where the chromosomes in the nucleus are replicated, separated, then divided into opposite ends of the cell in preparation for the cell to split.
Mitosis can be divided into four separate stages. Prophase, the first stage is where the chromosomes are replicated. In metaphase, the second stage, the chromatids (chromosome copies) are lined up along the equator of the cell. In anaphase, the chromatids are drawn to opposite ends of the cell, along spindle fibers, which are attached to centriole organelles, at the opposite ends of the cell. Finally, in telophase, the chromosomes are enveloped by the formation of a new nuclear membrane. Cytokinesis, mentioned earlier, occurs next , where the cell splits into two daughter cells.
Without these processes, life would not exist as we know it on Earth. For any species to survive, one of the most important requirements that has to be filled is the ability to reproduce, to propagate the species.