Each process is the replication of cells. Mitosis and meosis both take place in multicellular, eukaryotic organisms whereas binary fission is for unicellular, prokaryotic organisms. Mitosis and binary fission are both asexual processes that result in two genetically identical cells and meosis is the part of sexual reproduction that yields four genetically different sperm cells, or one egg and three polar bodies.
The purpose of mitosis is to replace dead or damaged cells in an organism. Mitosis is a small part of the cell cycle, or the life span of a cell. During interphase the DNA in the nucleus replicates. The first step of mitosis is prophase where the DNA condenses into chromatin the chromosomes are termed tetrads (tetra referring to the four legs on the tetrad) and the nucleus disintegrates, next the tetrads line up in the center of the cell during metaphase, thirdly in anaphase the tetrads are pulled apart and move towards the poles of the cell by spindle fibers that are attached to the centrioles, next the two new nuclei form with exact copies of the DNA in each nucleus during telophase and finaly cytokinesis (cell division) occurs. The result is two genetically identical diploid cells. Diploid meaning it has a full set of chromosomes.
Meosis occurs to create gametes or sex cells for sexual reproducing organisms. Meosis consists of two rounds of mitosis, however, in between the first and second round "crossing over" occurs where the genetic material is jumbled. Segments of the chromosomes cross with each other to change the genetic code leading to genetic variation. This process takes place so that offspring are not genetic clones of their parents. The result of meosis is 4 genetically different haploid sperm cells, or one egg with three genetically different haploid polar bodies (cells that provide nutrients for the egg). Haploid cells, or gametes contain half the number of chromosomes so that when the egg is fertilized the resulting embryo has a full set of chromosomes, half from each parent.
Binary fission occurs in prokaryotic cells, or bacteria. This is a very simplistic process compared to meosis and mitosis. This is an asexual process where the DNA that is just floating in the cell makes a copy of itself and the cell splits in two with one set of DNA in each cell. The purpose of binary fission is for rapid growth of single celled organisms.