Oh dear. So, first things first, it rather depends. Although the system remained more or less the same (although after the loss of Messenia and the subsequent liberation of the helots there was a great deal of administrative chaos), the relative power of various groups tended to fluctuate wildly. I wrote a comment here that, while it may not really address your question is still of interest, in that much of what is generally known about Spartan history and society is highly distorted by the Spartan Mirage.

Alright, so we know from Aristotle, who has somewhat mixed feelings about it, that the Spartans had a system in which two hereditary kings were backed up by a council of five ephors, or elders. The dual kingship may seem strange, but it's really quite common among late Dark Age to early Archaic Period states, since as cities began to form from the union of neighboring villages each community was represented by their own official. The "city" of Sparta (which is in quotation marks because it never fully unified, to the extent that well into the Hellenistic Period Spartiates were still referring to themselves as citizens of their respective villages, rather than the city of Sparta) was formed of about three villages, two of which were powerful and important. There was no infrastructure to speak of between them and they essentially existed independently from one another, with only the loosest ties connecting them. The kings are often mistaken for the officials, and sometimes compared by people who don't know any better to the executive branch of some modern constitutions. They are nothing of the sort. The kings held two functions, first as ritual leaders (many of the rituals in various Greek societies needed to be performed by a king--hence the position of the archon basileus in Athens, which was akin to the office of Pontifex Maximus at Rome, which also descended through the ancient kingship), and secondly as military leaders. That's it. That's all the kings are there for, to perform some sacrifices, preside over certain rituals, and lead the armies.

The real power of the Spartan state lay in the hands of the five ephors. These guys were theoretically peers, but in reality they squabbled so much that the most powerful or oldest (often the same guy) usually ended up in charge, with all the others just backing him up. The power of the ephors is difficult to exaggerate, and it's something that is often overlooked, especially because later scholars (using Aristotle as a model) liked to make a big deal of how stable the Spartan constitution was. It was stable, true. But only because the five ephors had total control over every aspect of the life of any subject of the Lacedaemonian state. The helots of course had no rights, and what rights they theoretically had could be revoked by any Spartiate at will. The ephors were in charge of the estates reserved for Spartiates at birth, but some time during the late Archaic Period more and more Spartiates began losing these estates, with the result being that a great deal of Spartiates amassed massive land-based wealth, although this was illegal. Many of these guys were ephors or their buddies, which indicates just how corrupt the body was. The ephors dictated every aspect of a Spartiate's life, since they legally could revoke any one of his rights temporarily or permanently. Now, as for the perioeci, who had nominal citizens' rights, the ephors still had control, although more indirectly. Since any Spartiate had the right to dictate orders to a perioeci on pain of punishment, and the Lacedaemonian state reserved the right to confiscate any land, the ephors were in charge here as well. The ephors were even in charge of their subject-allies' cities, since a condition of the Peloponnesian League was that the Spartans reserved the right to meddle in their allies' affairs, by setting up puppet states in the name of defending the old order. After all, since the Spartans did not record any administrative proceedings, it was the duty of the ephors to decide what the laws said.

Now, there's actually a third council in the government in addition to these two. There was a council of 30 elders (28 guys over 60 and the two kings, who could be under 60. The kings didn't actually have any power here, but merely presided in a ritual fashion) that had been set up by Lycurgus and was theoretically a balance on the power of the ephors, for which Plutarch praises Lycurgus. Aristotle, however, is very damning of this council. While Plutarch is far too willing to accept the supposed democracy of the elections within the Gerousia, and the pretense of fair voting, Aristotle makes it very clear that it was no such thing. Aristotle points out that the Gerousia was composed almost entirely of members of a handful of prominent families, with the two royal houses (the Argiads and the Eurypontids) forming the most important section of the Gerousia. The Gerousia's function was to prepare motions for the Apella, the council of Spartiates as a whole, to vote on. The Apella doesn't even deserve to get a real mention here, since anything that it decided on could--and would--be instantly vetoed by the higher-ups. The Gerousia reserved the right to veto anything passed in the Apella (and the ephors reserved the right to ignore the Gerousia, or even replace them if necessary by demoting them from Spartiate status--the theory that the Gerousia was to be the highest order of the government ignores the fact that the ephors could simply nullify anything by demoting you from Spartiate status, thereby forcing re-elections that they, being the most influential, could control, something that Aristotle comments on). The Gerousia also, at least in theory, was the ultimate law-court, where cases of murder and so forth were tried, even holding the right to try the kings. The problem is that, along with the constant threat of disenfranchisement by the ephors, the Gerousia's decisions here could be completely overruled by the ephors, who had the power to grant pardons. Real powerful body, wasn't it?

Now, the supreme power of the ephors was not constant throughout history. The version depicted here is of the ephors at their height, during the late 5th Century up until the late 4th Century, when Aristotle commented on the Spartan constitution. During the Archaic Period the Gerousia had slightly more power, especially since the ephors had not quite learned how to take full advantage of their power. But around the time of the Persian Wars the ephors began to abuse their power and essentially seized control of an already totalitarian state. Now, I mentioned the loss of the helots and Messenia in the middle of the 4th Century. This was a major blow, since the entire basis for the system was a hierarchy of authority, which eventually converged on an incredibly few. So with the whole base of the pyramid removed, and with the Spartiate class rapidly declining in numbers, the system was thrown into total chaos. Because of this, during the Hellenistic Period there were several attempts by the kings to reinstate their power over the ephors, most notably and most successfully by Cleomenes III, who was nevertheless unable to save the state.

Also, what I've written is awfully damning, isn't it? It's mainly based on what Aristotle says, and Aristotle is probably our best source here. Plutarch, however, is rather fond of the Spartan constitution (although he bases what he says on stories and a very selective analysis of Aristotle, filtering out all the unpleasant things that Aristotle says). The Spartans were ultra-conservative for most of their history (until Cleomenes III, really), and their system was set up along those lines. It was inflexible and did not easily adapt--in fact, it was designed consciously not to adapt. But, provided that the conditions were met for its continued existence (that is, provided that the state was not met with unexpected circumstances) it was highly stable. What Plutarch doesn't like to mention (besides the unpleasant fact that the system was completely totalitarian, which Aristotle harps on at length) is that when the Lacedaemonian state was confronted with the unexpected the system fell apart completely and was totally unable of dealing with the problem. This happened most notably in the Peloponnesian War, in the period of Spartan hegemony following (which lasted only a pathetic decade, as opposed to the fifty-to-a hundred years of Athenian hegemony, and the twenty or thirty years of Theban hegemony--mainly due to Spartan excesses, corruption, and tyranny, which drove even their closest allies to revolt), and after the loss of Messenia.