Short answers, pulled from the story.
The French Consulate was the top-level government of the First French Republic, lasting from the 9th of November 1799 to the 18th of May 1804. It replaced the Directory after the coup of 18 Brumaire and ended when Napoleon declared the French Empire.
Napoleon seized power through the coup of 18 Brumaire on the 9th of November 1799, forcing the sitting directors to resign. A remnant of the Council of Ancients then abolished the old constitution and ordained the consulate. A public referendum on the 7th of February 1800 confirmed the new constitution with a claimed 99.9 percent approval.
The Concordat of 1801 was an agreement between Napoleon and the Catholic Church, drawn up to serve Napoleon's political interests rather than the Church's. It satisfied the religious feeling of the French populace, neutralised the constitutional democratic Church, and stripped royalists of their strongest appeal to ordinary believers.
The Duke of Enghien was arrested after a royalist conspiracy to assassinate Napoleon and restore the Bourbon monarchy was uncovered in early 1804. At his military trial at Vincennes, he admitted receiving 4,200 pounds per year from Britain and waiting on the Rhine to play a role in any regime change. He was found guilty under a law of the 6th of October 1791 and executed in the ditch of the Vincennes fortress.
During the Consulate, Napoleon created the Légion d'honneur and the Concordat, consolidated national institutions including universities, a judiciary, banking systems, and a professional bureaucracy, and restored indirect taxes. The historian Robert B. Holtman described this period as "one of the most important periods of all French history" because of how long these institutions endured.
The Peace of Amiens with Britain, concluded in March 1802, gave Napoleon the political pretext to request a lifelong consulship as a reward from the nation. A referendum on the 2nd of August 1802 returned a claimed 99.7 percent approval, and the Constitution of the Year X dated the 4th of August 1802 formally confirmed him as First Consul for Life.