Chromosomes contain linear molecules of DNA, which encode the inherited genetic information. The cells of the body contain one set of chromosomes from one parent and another set from the other parent, a total of 46 chromosomes in humans. When the sperm and egg cells are formed they each contain only 23 chromosomes. This halving of chromosome number is due to a special type of cell division known as meiosis. During meiosis each paternal chromosome lines up with each maternal chromosome. These paired chromosomes are homologous, and contain DNA which is almost the same. Then a process known as genetic recombination occurs. In brief, the chromosomes break and rejoin, one or a few times along their length at random positions. Thus, the chromosomes in the sperm and the precursor of the egg have re-assorted DNA, and every one of these cells is genetically distinct. How does genetic recombination occur?
The information in DNA depends on the sequence of four units: adenine, guanine, cytosine and thymine, collectively known as bases, abbreviated to A, G, C, and T. Each DNA molecule contains two strands which are complementary to each other, such that G is always opposite C, and A is always opposite T. This is a major feature of the famous Watson and Crick structure of DNA. Recombination between homologous DNA molecules depends on breakage and the unravelling of strands. Then these DNA strands become paired with opposite partners, following the rule G opposite C and A opposite T. The structure formed is shown in the following diagram:

Holliday Structure
(The DNA strands are of opposite polarity shown by the thick and thin strands. The short lines are G-C and A-T base pairs)
There are three possible configurations, and it is now thought that specific proteins which interact with DNA stabilise the structure as a four way junction, as shown in the centre. The structures are converted to new linear DNA molecules following the breakage and rejoining of single strands. These new molecules have some DNA which originally came from the maternal parent and some from the paternal parent, within recombinant chromosomes.
The configurations in the diagram are known as the 'Holliday structure' or 'Holliday junction.' Its existence was deduced from particular genetic studies in several different fungi in the late 1950s, and the formal proposal was published in 1964. There is another important feature of the structure. Although homologous chromosomes have almost the same DNA sequences of A, G, C, and T, there are also a few differences. This means that when they interact, as in the diagram, occasionally A may be opposite A, G or a C, G may be opposite G or T, C may be opposite a C or T, and T may be opposite T. These are mismatched pairs, and there are eight possible mispairs. There is a DNA repair system which recognises these mismatches and restores normal Watson-Crick complementary pairing. This is also part of the process of genetic recombination.
Historical overview:
Holliday, R. The history of the DNA heteroduplex. BioEssays, 12, 133-142 (1990).