The typical Classical Greek phalanx consisted of all the hoplites in an army, massed together in a grid pattern of ranks (width) and files (depth). Most battle lines were made up of separate contingents from allied city-states, deployed side by side. One of these would form the right end of the line, another the left. Any additional contingents would make up the centre of the line. Even if the entire phalanx consisted only of the hoplites of a single city-state, there was usually still a low level of subdivision, so that a particular group of a few hundred men from one tribe or district would hold the right wing and another the left.

The flanks, then, would simply look like the extreme ends of the grid pattern. This was the point where the army ran out of hoplites to cover more ground. The deployment of the phalanx was always a trade-off between width and depth:

If we advance formed up many ranks deep, the enemy will outflank us, using their outflanking wing as they like; on the other hand, if we are formed up a few ranks deep, we should not be surprised if our phalanx is cut through by a hail of missiles and men falling upon us; and if this happens anywhere, it will be bad for the whole phalanx.

-- Xenophon, Anabasis 4.8.11

If a commander chose to deploy deep, to make his line more resilient, the men on his flanks would know they might be surrounded. On the other hand, if he chose to deploy wide, to prevent encirclement, the men all along the line would know that the enemy might break through easily anywhere, and encircle them from within.

With this in mind, you can imagine why Greek generals tended to prefer a deep deployment over a wide one. A wide deployment would make all the hoplites equally afraid; a deep deployment would only make the men on the flanks afraid. The trick was to put the toughest, most reliable contingents on the flanks, so that you could have the advantage of depth without having the fear of encirclement cause a panic at the extreme right or left. As long as the men on the flanks wouldn't give in to their fear, the battle could be won. This is one reason why Greek generals almost always led their armies from the right or left wing, rather than staying in the centre, as Persian generals usually did.

Another reason was the fact that every phalanx tended to creep to the right as it advanced:

All armies are alike in this: on going into action they get forced out rather on their right wing, and one and the other overlap with this their enemy's left; because fear makes each man do his best to shelter his unarmed side with the shield of the man next him on the right, thinking that the closer the shields are locked together the better will he be protected. The man primarily responsible for this is the first upon the right wing, who is always striving to withdraw from the enemy his unarmed side; and the same apprehension makes the rest follow him.

-- Thucydides 5.71.1

Because the men at the far end of the line were afraid of being encircled, and afraid of being attacked on their right-hand side where they had no protection from their shield, they steadily drew the entire phalanx off to the right. The result of this so-called rightward drift was that two battle lines that were deployed parallel to each other would be completely misaligned by the time they met in hand-to-hand combat. Each side's right wing would outflank the enemy's left. Each side's left wing would be dangerously outflanked by the enemy's right.

Greek generals understandably tried to do something about this natural rightward drift. One thing they could do was to place themselves and their best troops on the left wing, to prevent that wing from losing heart and to face the encircling enemy right wing head-on. Another thing they could do was to place themselves and their best troops on the right wing, and lead the outflanking end of the line around to attack the enemy in the flank.

The combination of these two factors meant that the battle was almost always decided on the flanks. Sometimes the contingents on the right of both armies would win, and the battle would end in a stalemate. At the battle of Koroneia, in 394 BC, both right wings won on their end of the line, after which the two victorious right wings turned around and battled it out with each other. At other times, though, the right wing of one side won by encircling the enemy left, and simply rolled up the enemy line from there.

Of course, Greek hoplites also tried to protect their flank in every conceivable way. They liked to deploy near terrain that would make outflanking impossible; a river or mountain on the wing to anchor the line was a fantastic way to prevent encirclement. They also used light infantry and cavalry to protect the flanks, and these were sometimes used to actually do the encircling themselves. Greek battles were never about hoplites only; often cavalry was only prevented from rolling up a battle line only because the enemy had brought its own horsemen to stop them.