He is played by John de Lancie, who does arrogant and self centred so very, very well.
Q is an omnipotent being from a race of other omnipotent beings also called the Q. When we first meet him he had taken members of the Enterprise crew hostage to use them as examples to judge humanity and to decide if he should banish them back to their little worlds.
He uses Farpoint station and what is going on there as a test. When Picard and his crew pass this test Q is not very happy, but he does abide by their agreement and leave them alone.
Of course he can't stay away forever. He had decided humanity is interesting, not so good for humanity, and decided to induct a human into the Q. He chooses Riker, first officer of the Enterprise, and if goes about as well as can be expected. He introduces Starfleet to the Borg, possibly speeding up the Borg's journey towards their sector of space. He is exiled and stripped of his powers and causes havoc on the Enterprise.
Over his exposure to humans he becomes less and less antagonistic to them, but he is always trouble. He even gets Captain Janeway and the crew of Voyage in the middle of a Q civil war, before ending the war by producing a new Q with a female Q and making Janeway his godmother.
I love Q. He started out a nasty, arrogant antagonist and grew into such a wonderful character. Somehow he maintained that arrogance without being a total and complete dick all the time.
Sources: My brain | Wikipedia
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