Short answers, pulled from the story.
The Pompeian Styles are four periods of ancient Roman mural painting, first identified by the German archaeologist August Mau. The four styles are structural, also called incrustation, architectural, ornamental, and intricate, running from the Roman Republic into the Imperial period.
August Mau, a German archaeologist who lived from 1840 to 1909, identified and described the Pompeian Styles from his study of wall paintings excavated at Pompeii.
The First Style, also called the structural or incrustation style, used stucco moldings and vivid color to simulate marble veneering. In Pompeii it dates from roughly 200 BC to 80 BC, though the technique existed elsewhere as early as the late 5th century BC.
The Second Style, or architectural style, replaced the marble illusion of the First Style with trompe l'oeil scenes that made walls appear to open onto imaginary landscapes and architecture. It dominated the 1st century BC and used relative perspective to push the picture plane back into the wall.
The eruption of Mount Vesuvius in the year 79 buried Pompeii in ash, and because the town was never reoccupied, the development of specifically Pompeian painting stopped there. The same four styles continued to be used and to develop elsewhere in Italy.
The Fourth Style, dated to around AD 60 to 79, is known as a Baroque reaction to the Third Style, reviving large scale narrative painting and panoramic vistas. The Ixion Room in the House of the Vettii in Pompeii survives as a prime example.