The last month of summer is a busy period in history, and probably every day has been significant in many years, though calendar differences can make it difficult to determine the date of an event. In this article, I plan to focus on key dates in 1812 in Russia, 1862-4 in the United States, and 1939 in Europe. Additional years may get added later.
To start with, the first major battle of Napoleon's invasion of Russia came at Smolensk on August 16-18, 1812. Like most of the campaign's battles, it was a hard slog (it didn't help Boney that the city was fortified). No precise count of casualties exists, but the estimates are that the French lost about 10,000 men and the Russians a little more. It was technically a French victory, in that the Russians under their best general, Michael Barclay de Tolly (odd name for a Russian), retreated toward Moscow, scorching the croplands behind him (after foraging them first). It wasn't much of a victory for La Grande Armée, though that is better than a defeat. But the Russians weren't too happy about Barclay de Tolly's retreats and his non-Slavic ancestry -- especially their other, more aggressive main army commander, General Pyotr Bagration of the Georgian royal dynasty of Bagrationi. Tsar Alexander I would replace him with Marshal Mikhail Kutuzov on August 27.
In 1863, William S. Rosecrans and Ambrose Burnside had already started out for Chattanooga and Knoxville respectively in mid-August. The first significant action came on August 21, when John T. Wilder's Lightning Brigade (mounted infantry armed -- through Wilder's own action -- with Spencer repeating rifles) began shelling Chattanooga. This helped keep Bragg's attention there rather than at Bridgeport, where Rosecrans made his main crossing.
The next important date is August 23, 1939. What is commonly called the Molotov-Ribbentrop Pact (officially a non-aggression pact) was signed on this day, probably guaranteeing an invasion of Poland. (Hitler wanted to invade it, but may have drawn back from it without such a deal and without Italian support. His only other "ally" was the puppet state of Slovakia.) This also enabled him to import badly needed resources, thus greatly mitigating the effects of the British blockade of the North Sea shipping lanes. Other raw materials were available from Italy, Hungary, Romania, and Yugoslavia, at least to some extent. In addition, they had a secret protocol dividing eastern Europe into spheres of interest. (This would be modified somewhat a month later.) Germany gave up more than Hitler would have liked, but it hardly mattered since he never planned on keeping the non-aggression pact any longer than he had to.
As just a minor addition here, the final offensive of the Second Boer War took place in August, culminating in a minor action at Bergendal on August 26-27, 1899. In a pattern that went back to the first significant battle of the war (Talana Hill), the Boers inflicted heavier losses but then retreated. The British quickly took the rest of the railway line to Lourenço Marquest, and formally annexed the South African Republic on September 1. After this the Boers resorted to guerrilla war before finally surrendering in 1903.
This was also when the Second Bull Run campaign was reaching its climax. Stonewall Jackson occupied Bristoe Station on John Pope's supply line on August 26. then sent Jeb Stuart's cavalry and an infantry brigade under Isaac R. Trimble on an overnight march to the fabulous Union supply base, which they captured the next morning. The VI Corps, having arrived in Alexandria by then, sent a New Jersey brigade under Nelson Taylor to find out why communications with the supply base had been broken. By then most of Jackson's men had arrived (except for Richard Ewell's division, minus Trimble's brigade, which had been left behind to delay Pope at Bristoe. Jackson then retreated back to a good defensive position while arriving the arrival of Lee with Longstreet's wing of the Army.
It's worth noting that Pope was hated not only by the Confederates (especially Lee, who said he needed to be suppressed) but by his own men. I believe it was Brigadier General John Hatch who told a colleague, "I don't care for John Pope one pinch of owl dung." His performance in the next few days would justify that hostility.