It's established Indians were the first to use elephants in war, as Indian elephants were more controllable than their African counterparts. Many Indian kingdoms used alcohol and other intoxicants to get the elephants high before sending them into battle, making them more pain-resistant and less prone to being terrified, and their riders were trained to sacrifice them with poisoned lances in case they got out of control anyway. A charge of intoxicated elephants with archers shooting down from their backs was pretty much enough to wreck any enemy formation. And to make matters worse for enemies, the Indians developed elephant armor, turning them into living tanks.
The Persians got their elephants and elephant trainers from India and frequently used the same tactics. Fifteen of them were meant to be used against Alexander the Great in the Battle of Gaugamela, and they caused such an impression that Alexander had to celebrate sacrifices to Phobos, the god of fear, the night before. Ironically, the Macedonians won the battle rather easily after the Persians ultimately decided not to deploy the elephants, as they deemed the beasts too tired to be used in the attack after they had used them to haul supplies (it's a matter to speculate whether having fresh elephants would have given the Persians victory, though, as the tactics used against the Persian war chariots might have probably been effective against them too). Still, after they captured the elephants, Alexander was so impressed with them that he added them to his own army. They proved useful when they began invading Pakistan and India, as his men already knew about them.
Alexander would finally face war elephants during the Battle of the Hydaspes against the Indian king Porus, who brought around 90 of them with tusks equipped with iron spikes. However, knowing he would probably lose an elephant-on-elephant battle and preferring finesse over force, Alexander only deployed his infantry and cavalry against Porus's army. He sustained heavy losses, but he ended up winning the battle thanks to his military genius: he ordered his men to loosen their ranks to allow the elephants to pass through and shower them with javelins and arrows, specifically targettting the mahouts so they could not sacrifice the beasts when they turned against them.
He would have advanced more towards India, but seeing that the next kings could deploy thousands of elephants against him, he wisely backed out. He founded an elephant guard in Babylon and created the post of "elephantarch" to lead them.
Alexander's general Seleucus used these to gain a decisive advantage over the other Macedonian generals in the Wars of the Successors, eventually conquering the lion's share of Alexander's empire minus Egypt. Elephants were used in many Hellenistic armies after that, and were helpful for instance in defeating the Galatians in Turkey in the 3rd century BC. However, after a while, professional soldiers got used to the sight of elephants, meaning their psychological impact was lost.
The Egyptian Ptolemaic kingdom (also descended from Alexander's conquests) and the city of Carthage (a Phoenician colony in Africa) also started using African elephants, as they discovered, probably from looking at the neighboring kingdoms of Numidia and Nubia, that there were local elephants they could use. However, they chose to take the smaller North African elephants rather than the huge savanna elephants, as African specimens proved to be much harder to tame than their Asian homologues (although it is known Ptolemaics and Carthaginians still traded abroad some Asian elephants). There is debate about whether those small elephants could still carry howdahs and turrets.
An elephant-on-elephant battle happened between the Hellenistic kingdoms of Ptolemaic Egypt and the Seleucid Empire at Raphia. However, the Ptolemaics' African elephants were scared away by the sight and smell of their larger Asian cousins, and the whole battle was a fiasco for them.
Carthaginians apparently only started thinking on war elephants after they saw those used by Pyrrhus of Epirus, who acquired a contingent of them from Ptolemy II in order to invade Rome. Ironically, although Pyrrhus managed to salvage two PyrrhicVictories because he had been the first invader to use elephants against Rome, the Romans adapted typically fast, and by the third battle they had learned how to kill the elephants or set them on their owners. Even although Pyrrhus was technically the victor, he never dared to show up in Italy again.
Carthage's use of elephants through the Punic Wars became stuff of legends, but in reality its success was similarly mixed, as by this point enough foes had already seen their fair share of trunks and tusks, so their usage now required some keen strategy and sense of chance rather than just sending them forward. They could still get away with it against the Gauls, who had never faced an elephant, but the Romans and Iberians were another story — Carthage's chief general Hamilcar died himself when the Iberian chieftain Orissus sent carts of burning wood against his elephants, making them panic and wreak havoc in his own camp. Hamilcar's son Hannibal Barca learned the lesson and became good enough to literally trample the Hispanics and Romans at several battles, managing to defeat and kill Orissus, but by the battle of Zama, the Roman anti-elephant measures had equally improved too much. At that place, Scipio Africanus left wide lanes open between his disciplined units, so that the elephants could take the line of least resistance by charging between them instead of over them, and also had some of his troops blow loud horns, which turned some of the elephants back on their masters. Nonetheless, Scipio took the elephants so seriously as a threat that he dictated, as part of the peace terms, that the Carthaginians should get rid of all their remaining elephants and not tame any more.
The Romans themselves adopted war elephants, although, perhaps because they were now so accustomed to neutralize them, they only used them occasionally, and still trusted more on their time tested formations and tactics.
A repeat of Raphia seemed to be coming between Rome and the Hellenistic Seleucids at Magnesia, where Roman general Lucius Scipio (Africanus' brother) brought 16 North African elephants against King Antiochus III's 54 AsÃan ones, but like Alexander in the Hydsaspes, the Roman ultimately chose not to deploy his beasts, aware of what had happened to the Ptolemaics. He instead utilized the usual anti-elephant measures and was victorious that way. Antiochus' use of his elephants was also frankly bad, placing them in spaces between his troops where they would cause most damage to his own ranks if they turned around (ironically, Antiochus had Hannibal himself as a consultant, but the king seemingly believed himself smarter and overruled him at any chance).
Rome also used the North African elephant in the conquest of Hispania, always as part of allied Numidian cavalry contingents, but they proved to be not too good for it, because many Hispanic peoples had previously fought either against or for Carthage and had passed on at least some second hand knowledge about elephants to their descendants (not to mention many of those had mercenary experience in Africa themselves). Roman war elephants marched at both the Celtiberian Wars and the Lusitanian Wars, but they achieved little success, and at the latter it even coincided with Rome's worst local defeat against the famous rebel Viriathus.
Special mention goes also to the Celtiberian city of Numantia. The sight of the Romans' ten elephants made the Numantines flee behind their walls, but when the Romans tried to tear down the gates of the city with the elephants, the defenders just threw rocks at them until a lucky shot injured an elephant on the head. The animal panicked, which in turn panicked the other elephants, and they all together trampled the Roman forces. The Numantines then made a sortie and killed three elephants and 4,000 men. This was such a catastrophe for the Romans that they called the day damned and never fought a battle on the same date, and the war against Numantia prolonged for almost ten years.
Some sources claim that the Romans brought one or two elephants along on their invasions of Britain, presumably on the basis that those natives, unlike the previous, would surely never have even heard of elephants, and so would be scared witless by the mere sight of them. It is unclear whether it was Julius Caesar or Claudius who was in command, but they apparently succeeded in scaring the Britons.
One of the last uses of elephants in Mediterranean warfare was against Julius Caesar at Thapsus. Specifically, they played a decisive role in saving Caesar from defeat, even though they were on the opposing side. Many of Caesar's legions were so sick of the civil war at that point that they charged without orders. With a sizable chunk of his army going rogue and making contact with the enemy, Caesar had no choice but to order the rest of his army to charge as well, which put him at a disadvantage against the already numerically-superior Republican army. This might have ended in total disaster for Caesar had it not been for the advancing enemy elephants. As the elephants charged Caesar's approaching legions, the legions created huge gaps within their ranks to let the elephants passed through, who then met with Caesar's 5th Legion. The 5th was specifically prepared with axes and trained to fight elephants, so they were ready to stop the beasts. While Caesar's army was fighting tooth and nails against the superior Republican army, the 5th Legion routed the elephants and sent them running back to the Republican army. Caesar's legions quickly got out of the way of the panicked beasts and allowed them to crash into the Republican's line, which completely devastated them. What followed was a complete massacre as Caesar's legions used the ensuing chaos to encircle the enemy legions and butchered everyone.
Medieval
The Umayyad Caliphate had been on the wrong end of war elephants during their invasions of Sindh, the first being foiled when the elephants' smell sent the Arab horses into panic. By the time of the second invasion, however, the Arabs had figured out that large volleys of flaming arrows would do the trick, and were able to crush the enemy army in open battle.
The Abbasid Caliph Harun al-Rashid gifted an elephant named Abul-Abbas to Charlemagne at the beginning of the 9th century. It is actually unclear if Abul-Abbas was trained for war or Charlemagne intended to use it as such, because later accounts were sensationalized. What is certain is that Abul-Abbas lived for eight years after arriving in Aachen, that Charlemagne took it with him when he marched north to answer a Danish attack on Friesland, and that the elephant died suddenly at some place beyond the Rhine called "Lippeham". The prevailing theory is that Lippeham is the confluence of the rivers Lippe and Rhine, and that Abul-Abbas died of pneumonia after being made to swim across the Rhine, which proved to be far too wide and cold for him.
When Timur the Lame invaded India, the Indians brought 120 armor-clad elephants with poisoned tusks against him. Timur ordered all his camels lit on fire and sent towards the elephants. The giant beasts turned around and trampled their own troops, thus winning the battle for Timur. He then had the same elephants incorporated into his own army, apparently thinking that he was the only one crazy enough to come up with a counter.
The people of Thailand have historically ridden elephants into battle. They are sacred animals there.
King Naresuan, who is still venerated in Thailand, is said to have fought and won a one-on-one duel with Maha Uparaja of Burma, with both on the backs of elephants wielding halberds (Naresuan's elephant had gone out of control and taken him too far into the Burmese lines; instead of retreating, he went for good old Bavarian Fire Drill and challenged the enemy leader to single combat). The duel is disputed by historians as only one official Thai source mentioned it with none of the Burmese and European ones mentioning the duel.
The King of Siam (Thailand's original name) famously offered Abraham Lincoln a herd of war elephants to help with The American Civil War. Lincoln politely declined King Mongkut's offer on the grounds that the States do not extend South enough to comfortably raise them.
The Burmese, Yunnanese, and Vietnamese all used war elephants in their wars against the Mongols, Chinese, British, and French among others. The Ming Chinese later took war elephants as tribute from Vietnam, even though the Ming never used them in war themselves.
The Khmer temples in Angkor Wat have several depictions of war elephants, including elephants carrying ballistae on their backs. It is unknown if these ballistae were shot from atop the elephants or they just transported them, however.
Genghis Khan was thoroughly unimpressed with war elephants, although to be fair, he never met them in their prime. When the Mongols marched on Samarkand, the locals rejected an open battle and stayed within the walls until they were made desperate by dwindling food and water. Then the garrison's weakened elephants charged at the Mongols but were forced to retreat inside when attacked by catapults. After the city fell, Genghis had the chance to add the elephants to his forces. When told that the elephants were hungry, he asked what they ate; when the Samarkandans replied that they ate vegetation, he said that there was plenty grass in the steppe and had them released. The elephants then died of thirst and starvation.
The Portuguese found war elephants used against them in the siege of Malacca during the Conquest of Portuguese India, but they overcame them through pike and shot.
King Mongkut's offer on the grounds that the States do not extend South enough to comfortably raise them.