The origin story
Even though the phalanx is probably most famous for being Greek, the earliest depicted phalanx formation actually comes from a Sumerian stele (a stone or wooden vertical slab, kind of like a totem) from the 25th century BCE. The soldiers there appear to have spears, helmets, and large shields covering the body, similar to the phalanx.
That isn't the only time that the history people were wrong. They used to think that the hoplite phalanx (which is the Greek phalanx that everyone knows and loves) came from 8th century BCE Sparta. Of course, we have no way of knowing for sure, but it's more likely that it came in the 7th century BCE after the invention of the aspis (a type of shield that the hoplon, the shield carried by the Greek phalanx soldier dudes, is based on) by the city of Argos. Without the shield that the other shield is based on, you couldn't possibly have had a Greek phalanx. This can also be seen in the Chigi vase, which shows the Greek soldier dudes with the aspis shields, spears and a set of equipment called panoply.
The Greek phalanx used 2 of the most basic principles of warfare, at least by those times' standards: take a large number of people, and make them work together. Over the years, as technology and weaponry developed, so did the phalanx, becoming more and more complex and effective, and eventually reaching as close to perfection as it could. Until it was drop-kicked by the Roman legion, but that's a story for later.
The Greek soldier dudes
These were hoplites, and they were what made up the phalanxes of ancient Greece. Hoplites weren't exactly a proper army, instead they were more like a militia, made up of common folk, with war as a part-time job. They weren't even provided with their armor or equipment, they had to buy and maintain it themselves, which caused a clear divide between who could afford to be in the hoplite army and who couldn't.
Individually, hoplites were weak. After all, they were pretty much just normal guys that were forced to go to war because their ruler wanted more land. They didn't have proper training or anything, just some basic instructions on how to not die immediately and sticking together in the phalanx formation. But, put a bunch of guys with armor and weapons into an uncomfortably close rectangular formation, and you get a recipe for victory. A recipe that lasted for a very long time.
The equipment
The equipment that the hoplites used was collectively called the panoply. It included:
A shield. But not just any shield, a special shield called a hoplon, which is what the hoplites were named after.
A helmet. This was usually decorated with a horsehair crest, bronze animal horns or ears, or paint. Which means it was entirely possible that someone had cat ear helmets in ancient Greece.
A cuirass, which is basically a breastplate and backplate strapped together. Most of the time this was just made of layers of linen or canvas stuck together to make it stiff, though some of the richer hoplites had ones made of bronze.
A pair of greaves, which are just plate armor that protects your lower legs. Like shin guards but heavier and more effective.
A double-edged, one-handed iron shortsword called a xiphos. Believe it or not, this was their secondary weapon, usually only used as a last resort if they had nothing else and needed to defend themselves.
A spear. This was the main weapon. The spear used in the Greek phalanxes was called a dory, and it was an 8-foot-long killing machine. The dory had an iron spearhead at the top, as well as a bronze spike at the bottom, so that the weight of the spear was perfectly balanced, as all things should be. It could also be used as yet another way to stab other people, just in case the pointy spear and the iron sword weren't enough.
By the way, all the bronze armor (the helmet, cuirass, and greaves) was really heavy, so much so that the soldiers didn't even wear it until just before they had to actually go fight. The helmet was usually tilted back on the head when not actively murdering people. It's similar to how you may put your sunglasses on your head when you're not in the sun, except with probably a lot more neck problems.
Why not just make a real army?
That is an excellent question. You see, back in the glory days of Greece, it was divided into a lot of city-states, who were kind of like siblings. Sometimes they were friendly (especially during they Olympics, they stopped entire wars for that), and the other 90% of the time they just fought each other.
Obviously, with that much war constantly going on, soldiers would run out quick, and they needed months or even years of training to be any good at the art of stabbing other people, so the kings decided to force the city-states' own citizens to fight, and so the hoplite was born.
Hoplites were special, because they didn't need a lot of training or expertise to go into battle, and phalanxes battles were over super quickly, which meant that the people could go back to their regularly scheduled lives pretty much the next day. Also, because neither side usually had much training, almost no one got badly injured or killed, since the only open spots in the panoply were really small targets, like the neck, and hard to hit even for someone who's had months of training. Sure, they took their hits of the spear, but they could walk that off in a few days.
In fact, the Greeks were so sure of their phalanxes, they didn't even make walls around their city-states until after the Persians rolled up, got obliterated by said phalanxes, and left.
Phalanxes in action
As with any army, the true potential of the phalanxes lay in the strength and discipline of the hoplites in them. The problem back then was that calling for backup didn't exist, and it wouldn't exist until the 5th century BCE, so a the outcome of a battle was decided solely on the people on the battlefield at that instance. And once the battle began, it wouldn't stop until one of the sides broke formation, and hence lost.
The hoplites would form the phalanx by standing in rows next to one another, covering the left parts of their bodies (and the right part of their left-side-neighbor's body) with their hoplons, and their spears sticking out from the gaps between the shields. Once in formation, the phalanx would move slowly toward the enemy, fighting them off slowly and blocking nearly all incoming hits with their shields, holding the formation as tightly as they possibly could in order to break through the ranks of the other side.
The phalanx was used on the front lines in a "pushing match" against the opposing side, quite literally since those in the front of the phalanx were often being pushed by the shields of those behind them. In later years, this strategy was used to make the phalanx as a sort of human battering ram to break into the enemy's front lines while the cavalry cleaned up the flanks.
The lineups
Over the years, there have been many different formations of phalanxes, each having better success rates than the last.
The phalanx at Marathon
This was the phalanx used by the Greeks at the famous Battle of Marathon against the Persians. This is the battle where the messenger ran for 26.2 miles from Marathon to tell the general that they had won against the Persians, and immediately collapsed afterwards from exhaustion and died on the spot. That story is the reason why running for 26.2 miles is called running a marathon.
The oblique phalanx
This phalanx was first seen in full force in the Battle of Leuctra in 371 BCE, when the commander Epaminondas of Thebes used it against Sparta, and wiped the floor with them. It works by placing the troops at an angle and focusing on one section of the enemy's battle lines at a time, eventually breaking through and winning.
Philip's phalanx
This was the phalanx used by King Philip II of Macedon (the father of Alexander the Great). In this phalanx, the hoplites had some change of equipment. They used longer spears, called sarissa (13-21 feet long) and smaller shields to help grip the spears more easily. Also, the phalanxes had a square formation of 256 hoplites, as compared to the rectangular formations of the phalanxes before them.
Hellenistic phalanx
This phalanx is a modification of Philip's phalanx, and it was used by Alexander the Great. This was arguably the most powerful of all the phalanxes that came before it, considering Alexander the Great was able to win most of his battles and conquer hundreds of kingdoms from the Mediterranean to Egypt and the northern coast of Africa and even as far as India using this formation.
The end of an era
Nothing lasts forever, and soon even the mighty phalanx was defeated. It began due to many weaknesses in the phalanx, and simply the development of stronger and more effective formations.
The main weakness of the phalanx was that its right side was almost completely open, because all the hoplites had their shields on their left arms. This apparently even caused them to drift toward the right, according to the historian Thucydides. Another important weakness was that phalanxes were only effective in a plain, since hills would break the line of battle, allowing enemies to enter and defeat the army. And finally, since hoplites never had any proper training, if the battle lasted too long, the frontmost line of soldiers would just collapse from being too tired, leaving the rest of the phalanx open and undefended.
This caused the phalanx to eventually lose out to the Roman legion. Their first battle was the Battle of Heraclea in 280 BCE, in which Pyrrhus of Epirus defeated the Romans, but lost a lot of soldiers because the Roman army was more flexible and could keep replacing the soldiers in the front line, which allowed them to keep fighting for much longer. The loss of King Philip V of Macedon at Cynoscephalae to Roman commander Titus Quinctus Flamininus in June 197 BCE is considered the best example of how legions are superior to the phalanx.
And so ended the glorious days of the phalanx.
Sources
History of the hoplite phalanx - qa.perl.org
What Was It Like To Fight In A Greek Phalanx? | by Erik Brown | History of Yesterday
The Greek Phalanx - World History Encyclopedia
Greek Battles: How They Fought, Hoplites & The Phalanx | HistoryExtra