Meiosis is a process of cell division that takes place only in sex organs, in order to produce sperm or egg cells. Sperm or egg cells must have half the number of chromosomes of normal, somatic cells, so that when they fuse during fertilization, the result is a zygote with the complete number of chromosomes. For example, human somatic cells have 46 chromosomes (diploid), whereas human sperm or egg cells have 23 chromosomes (haploid). To achieve this halving of chromosome number, meiosis is required.

Like in mitosis, cells are in interphase before meiosis begins, during which DNA replication occurs. Thus meiosis begins with DNA in the form of sister chromatids (see The difference between chromatin, chromatid and chromosome). Meiosis then takes places in two major phases: meiosis I and meiosis II.

Meiosis I
At the start of meiosis I (prophase I), homologous chromosomes find each other and pair up in the cell. They then exchange bits of DNA segments, a process called crossing over. Crossing over allows for greater genetic variation, as the resulting chromosome will have some genes from the maternal chromosome and some genes from the paternal homologous chromosome. At metaphase I, the homologous chromosome pairs line up in the equator of the spindle. Then, at anaphase I, the pairs are separated and moved to opposite poles. The cell then undergoes cytokinesis to divide into two daughter cells, the chromosomes decondense into chromatin strands, and the nuclear membrane reforms around the chromatin (telophase I). This means that at the end of meiosis I, the resulting cells will already have half the amount of chromosomes. However, a second phase is needed to separate the sister chromatids that form each chromosome.

Meiosis II
Meiosis II is very similar to mitosis. The chromosomes (sister chromatids) are separated into two individual chromatid strands so that there are 23 chromatids per daughter cell.

Thus at the end of meiosis, 4 haploid cells (gametes) are produced.

This image from the Molecular Biology of the Cell (Alberts, Garland Science 2002) shows clearly the steps...