Short answers, pulled from the story.
The War of the Second Coalition was a conflict fought from 1798 to 1802 between revolutionary France and a coalition of European monarchies that included Britain, Austria, Russia, the Ottoman Empire, Portugal, Naples, and several German states. It was the second major war between France and its neighbors since the French Revolution began.
The coalition failed largely because its three main partners had incompatible war aims. Britain and Russia sought to overthrow the French Republic and restore the Bourbon monarchy, while Austria, crippled by debt and internal instability from the First Coalition, prioritized strengthening its own position rather than invading France outright. Historian Paul W. Schroeder argued in 1987 that each major allied power was also scheming to limit the gains of the others, making sustained coordinated strategy impossible.
Napoleon played a decisive role in the war's second half. After abandoning his stranded army in Egypt in October 1799, he returned to France, staged a coup that made him First Consul, and took command of the war effort. He crossed the Alps through the Great St. Bernard Pass in May 1800, narrowly defeated the Austrians at the Battle of Marengo, and sent General Moreau to win at Hohenlinden in Germany, forcing Austria to seek terms by the end of 1800.
Austria signed the Treaty of Lunéville on the 9th of February 1801, accepting terms that closely matched the earlier Treaty of Campo Formio. France retained all its previous territorial gains and acquired new lands in Tuscany. Austria received Venetia and the former Venetian Dalmatia in compensation.
Russia left the coalition after a combination of military reverses and a diplomatic dispute with Britain. An Austro-Russian army was routed at the Second Battle of Zurich in 1799, and a separate Anglo-Russian invasion of the Netherlands ended in retreat at Castricum. Britain's insistence on searching neutral shipping in the Baltic Sea also alienated Tsar Paul I, who had already been angered by Napoleon's seizure of Malta and the expulsion of the Knights of Malta, of whom Paul was the honorary head.
The Treaty of Amiens, signed on the 25th of March 1802, ended Britain's participation in the war. It confirmed French control of the left bank of the Rhine and recognized the independence of the Cisalpine, Batavian, and Helvetic republics. The peace lasted only until May 1803, when Britain declared war on France again, beginning the Napoleonic Wars.