In this episode, the Enterprise, on its way to Farpoint Station to pick up its first officer and conduct some discreet investigations, is stopped by a powerful being that calls itself Q, and warned to turn back. Humanity, Q declares, is a savage race, and has expanded quite far enough into space already. Picard, unwilling to return to Earth at the say-so of Q, refuses, and he, along with several members of his crew, is put on trial for the past crimes of humanity. At length, Q sets a test for them, as Picard suggests, and they are sent on their way. The crew of the Enterprise eventually discover that Farpoint Station was created by exploiting an alien creature, which they free, passing Q's test.

Overthinking it

Trek is full of philosophical content, I always say. But "because I say so" isn't a good reason to believe it, so I'll take the opportunity to list out a few of the questions raised by this episode.

Inherited Responsibility

Q indicates that the people of Earth, as represented by Picard and crew, are responsible for the past transgressions of their ancestors. Is this true? Is it right to hold a child responsible for the sins of his father?

Collective Responsibility

More generally, do people have some collective responsibility for actions taken by others? Do you, as a human, have any responsibility for the actions of any other human, chosen at random? Do you have any responsibility for the actions of your government, or other representatives? For the actions of your family members? Members of a club you belong to?

Altruism

The Bandi caused the station to be built by a jellyfish-like alien they discovered on their planet, injured and in need of energy. They gave it enough energy to keep it alive, in return for it creating the station.

Did the Bandi have a moral obligation to help the jellyfish alien at all? They gave it enough energy to sustain its life. Did they have an obligation to give it enough that it could leave the planet?

Were the Bandi wrong to require the jellyfish alien to work for them in exchange for the energy they gave it? Why or why not?

The Bandi were in a position of power over the alien--a superior bargaining position. How does this affect their moral obligations, if at all?

Authority

Q acts as judge, jury, and prosecutor (and, potentially, executioner) for the crew of the Enterprise. He acts as though he has the authority to prosecute them for the crimes he says they have committed.

Does Q have any authority to prosecute the humans for their crimes, assuming they have committed any? If not, who does? If so, why Q?

Who has the authority to set rules that others should follow? As a citizen of a country on Earth, is it right that you are subject to the laws of the country in which you were born, even if you have not given your consent to them? If some governing group of aliens appeared and declared that as citizens of their galaxy, you were subject to their rules, would they be right?