In order to understand the success of Alexander, you must understand the success of his father, Phillip.
Phillip took a small backwater country, Macedon, and turned it into the de facto leader of all of the Greek city states. How did he do this?
First off, he modernized his army. The Macedonian phalangite used an 18 foot spear (the sarissa), wielded with two hands, with the soldier's shield slung on his left shoulder. The sarissa was double the length of contemporary Greek hoplite spears (which were typically about 8 feet long). The length advantage was important: the phalangite could engage with his weapon steps before his opponent could bring his spear to bear; in addition, the length of the sarissa enabled more spear points to be brought to bear against the enemy.
He also employ significant use of cavalry. While cavalry was used by other Greek states at the time, Macedonians were known for their horse breeding and riding expertise, and Phillip used that expertise to employ cavalry instead of infantry as the decisive element in his army.
Second, he developed tactics that were extremely effective against the current form of hoplite warfare in Greece. Before and during Phillip's time, a Greek battle consisted of multiple ranks of hoplites shoving against each other until one side became exhausted and routed (note: this is extremely simplified, I can elaborate on this if it isn't clear). Phillip did things differently. This is what the typical Macedonian battle order looked like. Phillip used his Phalanx as a holding force to lock the enemy infantry into a melee, while his cavalry and hypaspists (elite infantry) broke through at a weak point and flanked the main phalanx.
Imagine a mass of men, focused on nothing but pushing the enemy back, suddenly assaulted from the sides or rear. A flanking charge of cavalry typically resulted in the break up of a Phalanx and a complete failure of the nerve of the men in that Phalanx.
These tactics allowed Phillip to subjugate the Greek states, even when they allied against him at Battle of Chaeronea.
Another, lesser known bit of strategy about the Macedonian Phalanx is that, unlike the hoplite Phalanx, it was broken up into companies, platoons and squads, affording its individual parts significantly more autonomy and mobility than a giant block of infantry.
When Phillip was assassinated and Alexander took the throne, he utilized similar tactics to his fathers. The tactics of the Sarissa Phalanx holding the line while cavalry and mobile infantry flanked and routed the enemy is why Alexander was so successful. It worked against every enemy of that era.
If you'd like I can go into more detail about the Macedonian army's performance against specific enemies.
The phalanx, the sarissa along with the tactics used all undoubtedly contributed to the success of Alexander's army. Perhaps more importantly were the leadership skills of Alexander and the motivation of the Macedonians. At the Battle of Gaugamela Alexander's use of the cavalry on the right flank may be more about leadership than tactics and stands as one of the great successful gambles in military history.
The account of Alexander's speech and behavior at the Mutiny of Opis saw a discontented army challenging their leader turned around to begging for his forgiveness. Alexander personally engaged in battle, at times to the dismay of his army who saw no need for him to personally risk himself, but the respect for him was immense. He treated his army well with the spoils of war and at a very human level it appears he was perceived as having a true interest in the welfare of his followers.
Some will argue that by taking on Persian dress and customs Alexander alienated his Macedonian followers and there is evidence of this. As a whole, however, the Macedonian army was willing to follow Alexander into battles where they were outnumbered and at a disadvantage, demonstrating a great motivation to trust and follow their leader.
It's probably worth broadening the scope of this discussion a bit. I agree with most of the things said so far, but they're a bit narrow. I know there are some serious experts on the Ancient world on this board and I'm a little surprised none has weighed in, so I'll put in my thoughts.
Alexander's army was successful at a range of different levels.
At the level of individual battles they benefited from:
It is the combination of cavalry and infantry that is particularly important. Greek heavy infantry was already highly effective, but their cavalry was poor. The combination of heavy cavalry and infantry multiplied their effectiveness, and gave much more flexibility.
A crucial element that no-one has mentioned is their skill at siege warfare. This was just as important to an Ancient army as their prowess on the field. Alexander avoided some sieges by simply catching the enemy before they could withdraw to safety. Conversely, at Tyre, he knew when and how to dig in for a very lengthy siege. As he took over the Persian empire, Alexander gained much of their siege-engineering capability, which I believe was considerably better than that of the Greeks.
At the campaign level, Alexander's forces regularly moved with much greater speed than their opponents anticipated. In modern parlance they got inside their opponent's decision making cycle. By the time the Persians or the Indians or whoever formulated a response to Alexander's army it was too late, and they were already doing something else. Some of this is down to Alexander's dynamic and front-line leadership. It is also down to the speed of his cavalry and, perhaps most of all, the hypaspists. In Arrian's accounts these guys are regularly setting off on rapid marches that keep the enemy off-balance.
Alexander was also able to play local forces off against one another much more effectively than they could divide his army, although as mentioned below he did have serious issues with political cohesion.
At the level of grand strategy, Braudel argues that the collapse of the great Persian Empire was inevitable once they were cut off from the Mediterranean seaboard and its trade. The Persians then gave Alexander a model that was easy to take over. Essentially this was light-touch governance, and it didn't matter a great deal to most people whether they were ruled by Persians or Greeks. That said, keeping order and preventing corruption behind the front-line was a persistent problem for Alexander.
A final thing that I haven't seen commented on is that Alexander's army must have picked up a great range of knowledge and specialisms during its campaigns that more parochial opponents lacked. A tribe feels they're protected by an 'un-climbable' cliff? Well Alexander's got a bunch of expert mountaineers who can show they're wrong. Once you've marched and fought across the known world, you'll probably have more surprises for the locals than they have for you.
