It features a cameo appearance by a 137-year old Doctor Leonard McCoy, thus starting a tradition that the first episode of each new Star Trek series includes an appearance by a character featured in a previous series.
The episode shows the 'first' official mission of the new Starship Enterprise, NCC-1707-D. Captain Jean-Luc Picard has only just arrived, and it is to go to Farpoint Station to pick up the rest of its crew, including First Officer William Riker, Doctor Beverly Crusher, and engineer Geordi LaForge.
En route, the ship is stopped by a being who identifies himself only as Q, who demands that the Enterprise, and indeed all humans, return forthwith to the Solar System and stay there. A mock trial for barbarity for the events of the last several thousand years ensues. Data and Picard manage to persuade Q to permit humans to prove their worth, and are let free.
Upon arriving at Farpoint (a newly built starbase made by the Bandi, a race looking for membership of the United Federation of Planets and the economic and trade help that would imply), the crew are picked up. Riker has noted that a number of coincidences have happened, where people have asked the air for something, and it turns up later. Deanna Troi, the Enterprise's empathic counsellor, senses discomfort from the planet.
A ship arrives suddenly, and starts attacking the city next to the station. It it deduced that the ship is alive, and is attempting to free its mate, which had landed weak and distressed on the planet and was enslaved by the Bandi, who made use of its ability to transmute energy into matter. The Enterprise heals the captive ship by aiming low-intensity phaser fire at it, and it recovers and takes off.
After this, Q reveals this was the test set by him, and that the humans have passed it - at least for now.
The episode is followed up in the other Q episodes, and specifically All Good Things, the final episode of the series.