What are the purposes of mitosis and meiosis?

The purpose of mitosis is cell regeneration, growth, and asexual reproduction. Mitosis functions to replace older cells in the body that are no longer required. This is important in skin cells for example, as they are continuously dividing and being replaced by newer cells in the body. Animals and plants rely on the process of cell division to grow, as new cells are continuously added to their bodies/structures. Mitosis permits organisms to multiply their cells, resulting in the growth from a single cell into a complex, multicellular organism. In some multicellular and in many unicellular eukaryotic organisms, asexual reproduction is dependent on mitosis since it forms a genetically identical duplicate cell of the parent cell. For example, amoebas and yeasts use  mitosis to reproduce asexually, as a single cell divides to produce identical offspring. 

The purpose of meiosis is to generate sex cells for sexual reproduction. Four genetically distinct haploid daughter cells containing half as many chromosomes as the parent cell are formed in the process. More specifically, as these cells divide, pairs of homologous chromosomes are randomly separated into daughter cells. This process is known as independent assortment, and ensures that different chromosomes are distributed independently of each other. These sex cells may join together through fertilization and produce a new, diploid organism. The separation of homologous chromosome pairs during meiosis 1 occurs independently of the separation of other pairs of homologs (randomly). Meiosis contributes to genetic diversity of species through the process of recombination. More specifically, during prophase I, homologous chromosomes align in pairs and exchange DNA material.