Chromosomes form tetrads during the prophase I phase of meiosis, according to Biology Online. Tetrads occur when synapsis within the cell causes four chromatids to form as part of a homologous pair of chromosomes.
The word "tetrad" refers to the structure, as there are four ends visible when the chromosomes reach this part of meiosis. All of this occurs after DNA recombines during homologous recombination, usually resulting in the chromosomes crossing over, which leads to the tetrad structure. The next stage after crossing over and pairing off is leptotene, which occurs when the chromosome fibers coil up and condense together.