The image of the ship in Plato’s Republic is a metaphor that is meant to show what is wrong with democracy. Socrates is portrayed in this work as being in favor of rule by a philosopher king and being opposed to the sort of democracy that existed in Athens at that time. In the metaphor of the ship, he explains why he is opposed to democracy.
In the metaphor of the ship, Socrates tells the story of a ship that has a captain who is really not competent. The captain is a bit deaf, can’t see very well, and knows very little about navigation. His crew competes to be given the right to steer the ship since the captain cannot. The problem is that the sailors who are competing to steer don’t know anything about navigation either. They are not going to be appointed to steer because of their abilities. Instead, they are going to get appointed because they are the best at drugging their captain and because they are the best at killing their rivals. Once they take the ship over, they run it very badly. They use all of its stores for their own benefit. They promote their friends and they have no use for the people who actually know about sailing and navigation.
This ship is, in Socrates’ mind, analogous to a city-state that has a democratic form of government. Socrates is arguing that leaders in a democratic government are not chosen because of their knowledge of how to run a government. Instead, they are chosen because they can fool the people who vote (the captain in the metaphor) and they can outdo their rivals in dirty tricks. Socrates says that these people will not know how to run the government. Instead, they will just use it to enrich themselves and their friends. Socrates says that this is not how government should be run. Instead, government should be run by a philosopher king. This would be analogous to the one sailor in the metaphor of the ship who actually knows how to navigate and to steer. This person is clearly the one who should be navigating and steering the ship in the analogy and the government in real life.
In this metaphor, then, Socrates is trying to get us to understand why he thinks that democracy is bad and why it would be better if city-states were run by philosopher kings who had more knowledge of how to run a government than everyone else.