Meiosis is a special type of cell division. Unlike mitosis, the way normal body cells divide, meiosis results in cells that only have half the usual number of chromosomes, one from each pair. For that reason, meiosis is often called reduction division. In the long run, meiosis increases genetic variation, in a way which will be explained later.
Sexual reproduction takes place when a sperm fertilizes an egg. The eggs and sperm are special cells called gametes, or sex cells. Gametes are haploid; they have only half the number of chromosomes as a normal body cell (called a somatic cell). Fertilization restores the chromosomes in body cells to the diploid number.
The basic number of chromosomes in the body cells of a species is called the somatic number and is designated 2n. Thus, in humans 2n = 46. We have 46 chromosomes. In the sex cells the chromosome number is n (humans: n = 23). So, in normal diploid organisms, chromosomes are present in two copies, one from each parent (23x2=46). The only exception are the sex chomosomes. In mammals, the female has two X chromosomes, and the male one X and one Y chromosome.
All eukaryotes that reproduce sexually use meiosis. This also includes single-celled organisms. Meiosis does not occur in archaea or bacteria, which reproduce via asexual processes such as binary fission.
The offspring gets a set of chromosomes from each parent so that, overall, half of its heredity comes from each parent. But the two sets of chromosomes are not identical with the parental chromosomes. This is because they are changed during the reduction division by a process called crossing-over.
This is two-fold:
As with shuffling a deck of cards, many different combinations of genes can be produced without a change (mutation) in any individual gene. This greatly increases the variety of the offspring, and the variety gives at least some of the offspring a better chance of surviving in difficult times.
The recombination (crossing-over) which takes place in meiosis may be the reason why sexual reproduction exists at all. 
Several quite large taxa (groups of organisms) use cyclical parthenogenesis. This is when several generations are born by virgin birth, and then a generation occurs with normal sexual reproduction. Examples include aphids, and cladocerans (small crustacea called water fleas). Aphids usually operate as follows: when the weather is good, and their plant hosts are at their best, they use parthenogenesis. At the end of the season, when the weather gets worse, they use sexual reproduction. This system of reproduction is called apogamy.
In parthenogenesis, the eggs contain only the mother's genetic material, and they are not fertilized. The egg cells may be produced either by meiosis or mitosis. When meiosis occurs, crossing-over produces a genetic fingerprint which differs somewhat from the mother's. So, the parthenogenetic greenfly offspring are not identical, and do show some genetic variation: some chromosome segments differ because of meiosis. Mitosis would produce identical offspring.
Amongst these parthenogenetic taxa are a number of species which have entirely abandoned sex.
A few eukaryote organisms have completely lost the ability for sexual reproduction, and so do not have meiosis. These include the Bdelloid rotifers, which only reproduce by parthenogenesis. Little or nothing is known about this loss of sexuality.
Meiosis can be divided into Meiosis I and Meiosis II.
Prophase I: The chromosomes become visible, the nuclear envelope disappears and the centrioles (located at the top and bottom of the nucleus) begin forming spindle fibres that envelope the chromosomes.
At this stage each chromosome is split into two sister chromatids, held together by the centromeres. The paired chromosomes now have four chromatids (2 sets of 'sisters') pressed together. Crossing over takes place between two of the non-sister chromatids; the other two remain uncrossed. The crossover results in the exchange of segments of each of the participating chromatids, DNA and associated chromatin protein. Genetically, the process is called recombination.
Metaphase I: The chromosomes line up along the equatorial (the central line) of the spindle fibres in homologous pairs.
Anaphase I: The chromosomes are divided so that there are equal amounts on either side of the cell. As there are 46 chromosomes in a human cell, 23 end up on either side.
(Cytokinesis, the division of cells into two, takes place. The cell divides.)
Telophase I: The two daughter cells are completely divided, a nucleic envelope forms and the chromosomes become less visible. There are 23 chromosomes in each of these cells.
The two cells prepare to divide again in a stage known as Interkinesis or Interphase II. Both of these cells will go through meiosis II.
Prophase II: The chromosomes become visible, the nuclear envelope disappears and the centrioles form the spindle fibres.
Metaphase II: The chromosomes line up along the middle line on the spindle fibres
Anaphase II: The chromosomes get split into its two chromatids. Chromatids are the two strands of DNA (deoxyribo-nucleic acid) that make up the chromosome. They are joined by a mid-way connection called a centromere.
(Cytokinesis takes place. The cell divides.)
Telophase II: The cells are completely divided. The nucleic envelope reforms and four new cells with different DNA are created.
In humans, there are certain conditions that are caused by a meiosis gone wrong. Examples are:
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