Across history, there have been lots of wars that have decided lots of world history. If nothing ever got accomplished through war, or if people didn't at least think it was going to accomplish anything, it wouldn't be such a big part of history.
And in those wars, there are battles that make up that war. Those often get things done as well. Kursk burned up so many German resources that from then on they were doomed to lose World War 2. Waterloo was the effective end of Napoleon. The Tet Offensive turned Walter Cronkite against the Vietnam War, and American public opinion went with him, and Lyndon Johnson knew it.
But none of them had the this-one's-for-all-the-marbles stakes of the Battle of Gaugamela, on October 1, 331 BC. Gaugamela was a little burg existing at the time; the battle is also sometimes called the Battle of Arbela for the city, the capital of Kurdistan, currently standing 60 miles east of the battle. One might actually be better off pegging it as about 15-20 miles north of Mosul.
The stakes? Nothing much, just the fate of Western and Eastern culture throughout the world from then on.
For the west, Alexander the Great of Macedonia and you've probably already figured out who's getting the W.
For the east, Darius III of Persia.
Alexander was looking for vengeance on Persia for their earlier having tormented Greece from 499-448 BC, and having killed Alexander's father, and by the time he got to Gaugamela, he'd already done more than his share of conquering. Darius was on his heels, just hoping to fend Alexander off. In an earlier battle at Issus (in southern Turkey, near the Syrian border), he'd lost but managed to slip away, and even though Alexander captured Darius' family at Issus, he wanted Darius himself.
Darius came into the battle with more troops than Alexander- Alexander had about 47,000, while Darius had anywhere from 53,000-92,000- but it's not like that mattered, as Alexander was vastly shorthanded at Issus and smacked Darius around there too. Darius knew exactly what Alexander was capable of doing to him, even shorthanded, and it utterly psyched him out. Especially since Alexander had spent the two years since Issus carving his way from south-central Turkey to north-central Iraq, by way of Gaza, and snapping up anything that happened to be in the way, a time during which Darius had tried to buy Alexander off with anything and everything- land, money, his daughter's hand. Maybe he could at least get his family back.
As far as Alexander was concerned, the whole of Asia was already his anyway, and if he wanted to marry Darius' daughter, he could just up and do it whether Darius liked it or not.
Darius had guessed that Alexander was going to cross the Tigris River at Mosul, set up a sentry there to intercept him after what was to be a tough river crossing, and housed the main army in Arbela.
Alexander did not cross at Mosul. He didn't want to deal with fording a fast-moving river and then fighting right afterward. He headed further north. Now Darius was forced to scramble, and semi-arbitrarily picked Gaugamela as a substitute battle site.
Bad move. (There probably wasn't a good move, but still.) He'd chosen a place where he was on a plain, and the Macedonians would be up in the hills where they could easily make out the Persian layout beforehand. When Alexander showed up, he chose to rest up for the night, do some recon, and fight in the morning.
Darius called for an all-night guard, just in case. Alexander was in his head and now his army would be pulling an all-nighter to prepare for a fresh Macedonian army to boot. Alexander was pulling an all-nighter too, but he was busy working out how he wanted to do this.
Assume a map divided into squares:
When they met, Darius was in Square 2 guarded by infantry, with Alexander in Square 5 and his second-in-command Parmenion over on Square 4. The Persians were in three lines, with the Macedonians winged back on both flanks so as to draw the Persian fire. After feeling each other out, Darius chose to concentrate on Square 4 and Parmenion's men. Alexander, originally in Square 5, headed over to 6 and took the Persians that sat in Square 3. The point was to draw as many Persians as possible off of Square 2, leaving as little defense as possible around Darius himself.
First out of the proceedings were the Persian chariots, who ran into the Macedonian lances. The first line of lances stepped aside. The elephants running the chariots ran into the gap and then stopped, not wanting to run into the lances deeper in. The first line closed the circle. No more worrying about chariots. The javelin-throwers had a field day.
Once enough strength had been siphoned out of the Square 2 defense, the Macedonians- who had been forming a giant wedge with the point poking through the top of Square 5- pressed through, with Alexander behind the point. The Persians were split in two. Darius ran like hell.
Alexander surely would have taken off in pursuit, but around that time he heard that Parmenion was in trouble back at Square 4, and he chose to help him out instead. A few Persians saw a gap in the Macedonian line and ran through it, but there wasn't much to be done with it except loot from the Macedonian camp, which wouldn't even make up for the looting Alexander would be doing as part of the spoils of victory (and was unsuccessful anyway, as the back-line Macedonians easily drove them back.)
And it was a victory. It's estimated that by the end, the Macedonians had lost about 500 men, compared with at least 40,000 Persians. The Persians were all over the place, with Alexander's bodyguards taking up the pursuit of Darius, soon joined by Alexander after he had bailed out Permenion. Among Darius' remaining forces was Bessus, a nobleman and relative. A relative who would kill Darius himself, taking the Persian throne. Or what was left of it. Most historians, while conceding that he would be the next-in-line, don't consider him a king of Persia.