Short answers, pulled from the story.
The Battle of Waterloo was fought on Sunday the 18th of June 1815, near the village of Waterloo in what was then the United Kingdom of the Netherlands, now Belgium, about 15 kilometres south of Brussels.
Napoleon I commanded the French Imperial Army. The British-led coalition forces were commanded by Field Marshal Arthur Wellesley, the Duke of Wellington. The Prussian army was under Field Marshal Blücher.
A combination of factors cost Napoleon the battle. Marshal Grouchy failed to intercept the Prussian army, which arrived in strength on the French right flank from around 16:30 onward. The Imperial Guard's final assault on the Anglo-allied line was repulsed. With the Prussians breaking through at Plancenoit threatening his line of retreat, the French army collapsed into rout.
Hougoumont was a fortified chateau on the right of Wellington's line that Napoleon ordered attacked early in the battle to draw off Wellington's reserves. French troops briefly broke through the north gate after Sous-Lieutenant Legros smashed it open with an axe, but the Guards shut the gate and killed every Frenchman trapped inside. Napoleon ultimately committed 33 battalions totalling around 14,000 troops to fighting around Hougoumont across the whole day.
The British captured two French regimental eagles during the battle. The Royal Dragoons took the eagle of the 105th Line regiment, and the Scots Greys seized the eagle of the 45th Line regiment during the charge of the British heavy cavalry brigades.
Waterloo was Napoleon's last battle and the decisive engagement of the Waterloo campaign. He abdicated four days after the defeat, and coalition forces entered Paris on the 7th of July 1815. The battle ended the First French Empire and his Hundred Days return from exile, and is described as the second-bloodiest single-day battle of the Napoleonic Wars, after Borodino.