Crossing over occurs during prophase I, and it is the exchange of genetic material between non-sister chromatids of homologous chromosomes. Recall during prophase 1, homologous chromosomes line up in pairs, gene-for-gene down their entire length, forming a configuration with four chromatids, known as a tetrad. At this point, the chromatids are very close to each other and some material from two chromatids switch chromosomes, that is, the material breaks off and reattaches at the same position on the homologous chromosome. This exchange of genetic material can happen many times within the same pair of homologous chromosomes, creating unique combinations of genes, as shown in the animation below. This process is also known as recombination.

Independent Assortment and Random Fertilization
In humans, there are over 8 million configurations in which the chromosomes can line up during metaphase I of meiosis. It is the specific processes of meiosis, resulting in four unique haploid cells, that result in these many combinations. This independent assortment, in which the chromosome inherited from either the father or mother can sort into any gamete, produces the potential for tremendous genetic variation. Together with random fertilization, more possibilities for genetic variation exist between any two people than the number of individuals alive today.
Sexual reproduction is the random fertilization of a gamete from the female using a gamete from the male. In humans, over 8 million (223) chromosome combinations exist in the production of gametes in both the male and female. A sperm cell. with over 8 million chromosome combinations, fertilizes an egg cell, which also has over 8 million chromosome combinations.
That is over 64 trillion unique combinations, not counting the unique combinations produced by crossing over. In other words, each human couple could produce a child with over 64 trillion unique chromosome combinations!
How does crossing over, independent assortment, and random fertilization lead to genetic variation. Short sentence

Crossing over, independent assortment, and random fertilization result in the exchange of genetic material, the sorting of chromosomes into gametes, and the combination of chromosomes during fertilization, leading to a high potential for genetic variation in offspring.