Hoplite & phalanx

The hoplite was a heavy spearman named after his equipment (hopla), of which the key component was the large round double-grip shield called the aspis. In pitched battles of the Classical period (5th and 4th centuries BC), hoplites fought together in a regular grid formation of ranks and files. This formation initially had no name, but in the 4th century BC, the historian and veteran Xenophon revived an old Homeric term to describe it, which was "phalanx". So the short answer is that the hoplite is the warrior and the phalanx is his battle formation.

That said, the hoplite came first, and seems to have existed as a standard warrior type for centuries before the first sign of a regular homogenous formation (that is, a battle line consisting only of hoplites, rather than a loose and fluid one with archers and javelin throwers mixed in). The phalanx represents the end point of a slow development toward greater numbers and organisation of the hoplite body. I wrote more about this here.

This means a hoplite doesn't need to be in a phalanx to function. The phalanx may be his most dependable deployment in pitched battle - where a single phalanx is more than the sum of its hoplite parts - but he doesn't need to deploy that way in order to be an effective fighter. Hoplites fought in many different situations (ambushes, siege assaults, street fighting, running battles, fighting on ships) where there was no room for them to draw up a fully formed phalanx. The fact that they were well-protected, aggressive spearmen made them useful in a lot of situations, even if they didn't have time or space or numbers to form a phalanx.

It's only in major battles that the phalanx was indispensable. A more fluid deployment would easily fall into chaos, and was vulnerable to infiltration by the enemy. Good order was essential to maintain an unbroken battle line. On at least one occasion, in 414 BC, Syracusan generals abandoned their plan to fight a battle because they couldn't get their hoplites into formation. The Spartans, who made more of an effort to establish officer hierarchies and drill their hoplites, had a noticeable edge in pitched battle because their phalanx was quicker to form and easier to maintain in its perfect order.

 

Hoplite & state

Greek states by and large did not have standing armies. They defended themselves by calling their adult male citizen population to arms when war broke out. Each citizen was required to bring whatever weapons he could afford, and on that basis he would be deployed as a light infantryman, a hoplite, or a horseman. All the hoplites, regardless of their social or economic background, would be deployed together (usually in regional units) to form the city's phalanx. From the fifth century BC onwards, they would be paid a daily compensation for the duration of their campaign.

In places like Athens, which had a substantial population of immigrants, non-citizens were also required to serve, and fleshed out the ranks of the phalanx. At Sparta, the rapid decline of citizen numbers throughout the Classical period meant that freeborn non-citizens called perioikoi eventually formed the majority of their hoplite line. For long-term deployment in distant lands, a levy of leisure-class citizens was often supplemented with mercenary hoplites, who would form their own units within the phalanx.

Even so, hoplite service was generally regarded as the archetypal military experience for Greek citizens, and as their principal obligation to their state. Public monuments commemorated hoplite achievements and the hoplite dead; poetry and art presented the hoplite as the ideal and standard citizen, even if most citizens would have been too poor to fight as hoplites. In court and assembly speeches, the audience is assumed to consist of hoplites even if this was blatantly untrue in practice. Hoplite service was a symbol of what it meant to be a citizen, and all citizens (unless they were rich enough to serve in the cavalry) aspired to take their place in their city's phalanx.

The phalanx, then, was the heavy infantry core of the citizen militia; it was one of the main ways for the state to protect its territory and project force; and it was the ideological keystone of what it meant to serve your community in war.

I hope this answers your questions! If you have more specific points you're wondering about, or if you'd like me to recommend some reading, feel free to ask.