Short answers, pulled from the story.
The Battle of Austerlitz was fought on the 2nd of December 1805 near the town of Austerlitz, known today as Slavkov u Brna in the Czech Republic, about ten kilometres southeast of the city of Brno.
The nickname comes from a letter Napoleon wrote to Josephine after the battle, in which he said he had beaten the Austro-Russian army commanded by the two emperors. Napoleon was the third emperor involved. He was mistaken about Emperor Francis of Austria being present on the battlefield, but the phrase stuck and became the battle's second name.
Around 158,000 troops were involved in total. The Allied army lost about 36,000 men, roughly 38 percent of its effective force of 89,000, along with around 180 guns and about 50 standards. French casualties were around 9,000 out of approximately 66,000 engaged, or about 13 percent.
Napoleon deliberately feigned weakness by abandoning the commanding Pratzen Heights and weakening his right flank, luring the Allies into attacking there with most of their force. Once the Allied centre on the Heights was emptied, Marshal Soult's IV Corps of 16,000 troops stormed up the slope through morning mist and demolished the Allied line, allowing the French to encircle both flanks.
The Treaty of Pressburg, signed on the 26th of December 1805, removed Austria from the war and imposed an indemnity of 40 million francs on the Habsburgs. Napoleon used the victory to create the Confederation of the Rhine, which led Emperor Francis to dissolve the Holy Roman Empire in 1806. Prussian alarm at French expansion triggered the War of the Fourth Coalition that same year.
The battlefield area has been a protected landscape monument zone since 1992, covering 19 municipalities. The Cairn of Peace Memorial near Prace, designed by Josef Fanta and dedicated in 1923, is claimed to be the first peace memorial in Europe. In Paris, the Colonne Vendôme, the Pont d'Austerlitz, and the Gare d'Austerlitz all carry the battle's name, and a relief depicting the battle appears on the Arc de Triomphe.