Socrates summarizes the decisions they have made about the city. Wives, children and their education must all be in common. The philosopher-kings are to be drawn from the best among them, live in common buildings, and excel in warfare and education. The other citizens will provide what the philosopher-kings need to live, so that they may rule.
This basic description of the city omits the importance of specialization into producers, auxiliaries, and guardians. It also omits the details regarding education, covered earlier.
Socrates describes the four types of government— Timocracy, Oligarchy, Democracy, and Tyranny. All are failures. He adds that there are equivalent kinds of human soul for each government. Socrates imagines a gradual failure of the city as it passes through each government. Because the city is human, it is imperfect and thus destined to fail. The future leaders are chosen based on human perceptions. Eventually, errors in selecting future guardians will result in someone who is not fit to rule being selected.
Socrates describes the failure of his ideal city as it passes through the four forms of government. Notice too that the failure begins in the first stage of the education process, because children who should not have been chosen as guardians were chosen.
Rulers, selected in error, will desire property. Eventually the rulers will own all the property. They will emphasize military training in order to control the producers. This will lead to Timocracy, in which the military rules, and honor and victory are valued above all else. Leaders will be chosen not for wisdom, but for their abilities in war. The Timocratic man's soul is controlled not by reason, but by his spirit.
In modern terms, the Timocratic man is controlled by his emotions.
In an Oligarchy all political power is in the hands of the wealthy. The wealthy and the poor are at war with each other, so that there are really two cities. The Oligarchic man is motivated only by greed. Eventually the Oligarchy increases poverty until the poor rebel.
The flaw of Oligarchy is an emphasis on wealth as the virtue, which Socrates thinks is really a vice. Because of extreme wealth and extreme poverty the city is essentially divided into two cites.
When the poor rebel against the wealthy minority the city becomes a democracy. In a Democratic city no one is forced to take public office, no one enforces law, and no one serves in the military. Eventually a leader rises because he panders to the people. As his popularity rises so does his power. Eventually those who lost wealth start a civil war. The Democratic man moves from momentary whim to momentary whim, without moderation or order in his life, and thus accomplishes little.
The flaws of the Oligarchy lead directly to Democracy. You will note that it is almost the worst form of government, in Socrates' view, surpassed in its flaws only by tyranny. The central flaw in democracy, in his view, is an uncontrolled desire for freedom, which evolves into anarchy, thus leading to tyranny an attempt to control the anarchy. During Plato's life, Athens was briefly controlled by an Oligarchy, which rapidly decayed into a democracy, which executed Socrates.