1. Two Kings. At the same time.
In 300, there's just Leonidas, but most of the time the Spartans actually had two kings in office. There were two separate royal lineages, the Agids, who were marginally more important, and the Eurypontids, and are supposedly derived from the legendary twins Eurysthenes and Procles.
This actually caused fewer problems than you might think, because the Spartan kings were not autonomous. There were several other groups who had control of things like law and government policy. The kings were more like permanent generals. Once, two kings (Cleomenes I and Dematatus) went on campaign together, had a girly spat about Athenian democracy, culminating in Dematatus going home early and taking the ball- or in this case, their helpful allies the Corinthians. Naturally, they couldn't be friends after this, and Cleomenes eventually schemed to have Demaratus overthrown. Demaratus then fled to Persia, where he advised Xerxes about the Spartans at Thermopylae. This incident culminated in legislation preventing them both from going on campaign at once.
Other powers of the kings were mostly army-based. They were head of their barracks, and allowed two of the allotted rations of food, though the implication was that this was to award to someone else, rather than something they got to keep for themselves. It's also not as cushy as it sounds: one visitor, after tasting their famous 'blood soup' remarked that he "now knew why the Spartans were so keen to die". They were also, crucially, allowed a personal body guard of up to 300 men. Which brings us to...
2. Extremely Pious
As Herodotus puts it, the commands of the gods were more important to the Spartans than the commands of mortals. While in 300 the Ephors are crazy, inbred, historically inaccurate monsters, that they forbid the war is actually correct. The Spartans in general were incredibly pious, always making certain to run their decisions by the Oracle at Delphi, and making sure to check the animal guts before they did anything.
One guy, Tisamenos, an Elean, went to the Oracle to ask 'how to get children'. I'm not sure exactly what he meant by that, but I like to think they were having fertility trouble, rather than that he just didn't know what to do. Instead of offering any helpful advice, she told him he would win five great contests. He assumed that he would win the pentathlon, and promptly entered. He only won four of the five events, so clearly this couldn't have been what the Oracle was talking about, which meant he was clearly going to win five battles. The Spartan nobility decided to try and engage him as their divine auger, to check the omens were right before they went into battle. They were so desperate to have him onside that he was able to secure full Spartan citizenship for himself and his brother, the only two foreigners known to have achieved this.
At the time when Leonidas wanted to leave Sparta for Thermopylae, it was the Carneia, one of the most important festivals of the calendar, and so (supposedly) the army wasn't allowed to march to war. Hence, Leonidas and his 300 bodyguards decided to "take a walk".
Just this once, that excuse might have worked, except that this was their excuse for not being at Marathon too. It's generally thought a lot more likely these days that they didn't like to leave the city for too long in case of a slave revolt.
3. Slave Population
They didn't show up at all in 300, but all Spartan warriors had an allotted number of Helots, who were a kind of mixture of serf and slave. Serfs in that they were allowed to keep some of their produce, and slaves in that they were owned and could be disposed of by their masters. Though almost all Greek cities kept actual slave populations, they had only enslaved foreigners, who were obviously naturally inferior, whereas Sparta enslaved fellow Greeks which was clearly not on. These slaves were the people of nearby villages and a neighbouring town, of Messenia.
Myron of Priene said that: "They assign to the Helots every shameful task leading to disgrace. For they ordained that each one of them must wear a dogskin cap and wrap himself in skins and receive a stipulated number of beatings every year regardless of any wrongdoing, so that they would never forget they were slaves. Moreover, if any exceeded the vigour proper to a slave's condition, they made death the penalty; and they allotted a punishment to those controlling them if they failed." and Plutarch adds that they were forced to drink neat wine until drunk, and then "lead them in that condition into their public halls, that the children might see what a sight a drunken man is; they made them to dance low dances, and sing ridiculous songs."
The numbers of these slaves and the number of Spartan warriors varied over time, but at the most extreme there were probably 10,000 Spartans and 250,000 Helots. Obviously, being this outnumbered, they had to take some quite bizarre measures in order to control them.
Myrion of Priene again: "The helots were invited by a proclamation to pick out those of their number who claimed to have most distinguished themselves against the enemy, in order that they might receive their freedom; the object being to test them, as it was thought that the first to claim their freedom would be the most high spirited and the most apt to rebel. As many as two thousand were selected accordingly, who crowned themselves and went round the temples, rejoicing in their new freedom. The Spartans, however, soon afterwards did away with them, and no one ever knew how each of them perished."
As you can imagine, the Spartans weren't too keen to take their army away from the city, when there could be a revolt in their absence. There were several very serious ones during Sparta's hey-day, including one that lasted ten years.
It is another attempt to control the Helots that forms out next section.
4. Roving Death Squads of Teenagers
You've probably heard all about the Spartan school system. If not, here's a handy explanatory video. Reports vary, but either in your last year in the system, known as the Agoge, or just afterwards, and in both cases were really good at being a Spartan, you were drafted into the Krypteia. This was essentially the Spartan secret police, who existed mostly to prevent Helot revot by killing them whenever and wherever they saw fit. They were presumably selected by the Boy Herd, who oversaw the whole Agoge. Being selected was a great honour, and usually earmarked you for one of the higher offices of state. Like many other Spartan practices, the Krypteia was all about becoming a better soldier, and, unexpectedly, this last point was designed to do just that.
5. Dance Champions of the Greek World
The Spartans were all about being the best at things. If that included singing and dancing in unison, well, fine then. According to Cartledge, ''The Spartans were famed for dancing in general, and for one particular military dance, the Pyrrhic''.
The reason the Spartans were so good at fighting was (as we learned from 300) that they were really, really disciplined at phalanx fighting. Co-ordination in phalanx formation is incredibly important, so all the dancing and singing in groups had two purposes: one, to help everyone keep in step when it really mattered (i.e. when fighting the Persians or the Athenians), and two, to keep the gods onside, because dancing and singing were also forms of cultic worship, and as we saw earlier, the Spartans were pretty damn pious.
You heard it here first guys. Unless you've read your Cartledge.