The thousand-year-old secret society depicted in "The Da Vinci Code" is the Priory of Sion.
In Dan Brown's story, the Priory of Sion is portrayed as a hidden European organization guarding a secret connected with Jesus, Mary Magdalene, the Holy Grail, and a possible bloodline. The novel links the group with famous historical figures, including Leonardo da Vinci, Isaac Newton, and Victor Hugo.
The society's alleged ancient origins are a major part of the novel's fictional mystery, but they should not be treated as established history. The modern Priory of Sion was founded in France in 1956 by Pierre Plantard, and documents claiming a medieval history were later exposed as fabrications.
Readers also commonly confuse the Priory of Sion with Opus Dei. In the novel, Opus Dei is depicted as a powerful Catholic organization opposing the Priory's secret, but it is not the answer to this question.