This is a complicated question as Spartan governance was created from multiple orders which balanced and shared power. Most of the information here is from ancient historians like Herodotus, Aristotle and Plutarch who wrote either after Sparta's collapse and/or were highly prejudiced.

This information was rather quickly gathered from some old notes I had and I hope is a permissible answer with the rules of this subreddit.

Kings:

Sparta was a ‘diarchy’, ruled by two kings who had religious, judicial, militaristic and political roles. The kings came from two royal families, the Agiad and the Eurypontids who claimed to be descendants of the god Heracles, thus the office of ‘king’ was hereditary.

Herodotus suggests they were given two priesthoods, Zeus Lakedaemom and Zeus Uranus. They had duty to perform all sacrifices on Sparta’s behalf (Xenophon) and were charged with safekeeping the oracles. Spartans regarded them as demi-gods, evidenced by the reverent funerary customs that were attended to them.

• Sacrifices included animal sacrifices to Apollo on the 1st and the 7th of the month. The kings wore the skins of the sacrificed animals and were the first to sit at dinners following sacrifices.

• Elected two Pythioi who consulted the Delphi Oracle on their behalf.

• Made the first ceremonial libation-drink offered to deity.

Military: Herodotus indicates the kings could declare war on whoever they pleased.

• One king accompanied army in battles as the commander in chief. Aristotle describes them as being ‘hereditary generals’. The king’s involvement in war is suggested by the King Leonidas’ bust.

• Accompanied by a bodyguard army of 100 picked men. (Herodotus)

• Whilst one was in war, the other tended the city.

Judicial: they were judges and law givers.

• Vowed with Ephors monthly to rule by Sparta’s laws.

• Overlooked adoption and marriage of orphaned heiresses.

• Sat on the Gerousia with 28 other members.

Ephorate:

5 ephors made up the ‘ephorate’, their origins were unknown but there were suggestions that they were possibly ancient priests/ astrologers, appointees to the king, tribal chiefs from the five villages who were advisors to the king. They were not mentioned in what we found were the earliest constitution documents and hence can only be concluded that ephors were set up later in time. Any Spartan male over 30 could be elected for the office, which lasted for one year and could not be re-elected. Their office residence was called the ephoreion, located in the market place.

Ephors appeared to have taken some powers from kings and gerousias. Aristotle stated they had ‘supreme authority in the most important matters.”

• Chief administrators and executives of the state, they summoned and presided over the gerousia and the assembly.

• Exchanged monthly oaths with the kings, attended to and kept the kings in check. They could recommend charges be brought against the king.

• Xenophon suggested they had the right to inflict punishments at will.

• Had power to depose magistrates during their term of office, to imprison them and put them on trial for their lives. –Xenophon.

• Controlled public finances (including booty after war), and kept the Krypteia under control.

• Were responsible for discipline and education of the young. Dealing with misconduct and supervision of physical examinations.

• Sat with the Gerousia as a court of criminal justice, were sole judges involved in criminal cases with perioecoi, judges in all civil cases.

• Dealt with foreign embassies and permission for foreign delegations to enter Sparta.

• Selected age groups to go to war. Selected the royal bodyguards, and gave orders to generals.

Gerousia – the council of elders

Comprised of 28 Gerontes (men over 60) and the two Spartan kings. Gerousias were elected for life, by the assembly (ekklesia). The office of Gerousia held great influence and prestige. Supposedly every Spartan male could be chosen for the office, but realistically the members were mainly chosen from a small circle of aristocrats and wealthy families. They were responsible for:

• Preparing business and agenda for the assembly including framing the wording of legislation.

• Wide judicial powers, including death, exile and disgrace.

• Were able to put the king on trial.

• Ignore the votes of the general ekklesia if they wished.

• They deliberated public policies.

Plutarch – “It was by shouting that they decided between the competitors (candidates for the gerousia) … whoever was met with the most shouting, and the loudest, was the man declared elected”

Aristotle- “those who have had a share in this office have manifestly been guilty of taking bribes and have been corrupt enough to give way a lot of public property.”

** The ekklesia** – the assembly of the people

Also referred to as the Apella, was the public assembly that was comprised of all male Spartiates over the age of 30. The ekklesia met once a month, outdoors, to discuss proposals from the gerousia. Thucydides suggested that they voted yes or no by shouting (acclamation) or physically dividing into groups. However, the ekklesia could not debate issues, change the wording of motion, start considerations of new issues or propose new laws.

• Plutarch reports that an addition clause, known as the rider, further limited the powers of the people by asserting that gerousias could ignore decisions of the ekklesia if they thought it unreasonable or unsuitable.

The ekklesia’s main roles and powers were:

• Appointment of military commanders, generals and admirals.

• Elect ephors (5 each year), and gerousias.

• Elect magistrates decisions concerning peace and wars, resolutions regarding kingship, emancipation of helots and voted on foreign policies.

Spartan systems aimed to achieve ‘eunomia’, meaning ‘good order’, emphasised by duty and obedience to the laws.

While Sparta was ruled by two kings, I think that people would classify there system as democratic. All citizens, or "Homoioi", were expected to participate in government, discuss issues, and participate in general assembly. Homoioi were the first-class citizens and were granted many rights. Second class citizens usually were there for labor and merchant trade. Anyways, the citizens of Sparta would vote on issues and elect a council of elders that gave advice and trained young soldiers, but the people made most of the decision making.

I would classify it as a direct democracy.

Was the assembly open to all Lacedameons, some groups, or only the Spartiate peers? Because if it was only open to Spartiate peers then I would say its an oligarchy.

Also according to John Buckler Sparta's foreign policy propped up oligarchies against democracies, while Athens did the opposite.