Daniele Barbaro was a member of the Venetian aristocracy (see Barbaro family). He was an important prelate, humanist and architectural theorist, who commissioned a number of works from Veronese (including a portrait in the Pitti Palace where Barbaro is dressed as a Venetian aristocrat). Here Barbaro is portrayed sitting in the "udience" posture (reserved normally for Popes and cardinals) in recognition of his status as cardinal and Patriarch of Aquileia.
The book standing up is the La Practica della Perspettiva, Barbaro's treatise on artistic perspective. The other volume on the table is Barbaro's "Commentary" on Vitruvius' De architectura, which has illustrations by Andrea Palladio. Veronese had been involved directly with Barbaro and his brother Marcantonio Barbaro, decorating the Villa Barbaro, Maser, which Palladio designed. Barbaro's Commentary on Vitruvius was published in Italian in 1556, but the portrait may be linked to the publication of a second edition in Latin in the 1560s.