“Ah,” said Pius Thicknesse. “Has he been caught having contact with an Undesirable?” -- Pius Thicknesse (DH13)
Pius Thicknesse was the head of the Department of Magical Law Enforcement (DH1) until the death of Rufus Scrimgeour, after which he became Minister for Magic (DH13). Unfortunately, he had been placed under the Imperius Curse by the Death Eater Yaxley earlier in the summer (DH1), which meant that from this point through the rest of the war, control of the Ministry was effectively in the hands of Voldemort.
During Thicknesse's Ministry it became a crime to be a Muggle-born wizard. The official persecution began with a Muggle-born Register, "to understand how they came to possess magical secrets," supposedly supported by new research that "proved" that Muggle-borns were "likely to have obtained magical power by theft or force." Neighbors and co-workers were exhorted to report Muggle-borns who had not registered with the Ministry, or who had faked their ancestry documents. A blackmarket industry in ancestry documents thrived, but many unfortunate wizards were stripped of their wands and cast out of Wizarding society or imprisoned at Azkaban (DH13). Muggle-born children were denied admittance to Hogwarts as well (DH11).
Thicknesse fought with the Death Eaters in the Battle of Hogwarts and was jinxed creatively by Percy Weasley who hollered as he fired off the spell, “Did I mention I’m resigning?” Percy and his father later attacked Thicknesse a second time during the battle, flooring him (DH36).
Other canon notes and references
Appearance: Thicknesse has long black hair and beard streaked with silver and a large, overhanging forehead (DH12).
First name meaning: To be 'pious' is to be noticeably and devoutly religious or obedient; today the word can sometimes imply hypocritical behavior. 'Pius' is an old Roman title used by emperors, and later, by Italian Popes.
Last name meaning: 'thick' is slang for stupidity.
Together his name would appear to mean 'pious stupidity,' an inauspicious name for a Minister of Magic. Reader Carol Braun noticed on the Midnight reading video [TLC] that Rowling pronounces the Minster's name as Thick-NESS.