The Q are undoubtedly one of the most interesting and peculiar species ever encountered.  Little is known of them for certain but they are known to reside in a place they refer to as 'the Q continuum', which is apparently a higher dimension of some form.  The Q are thought to have once been a relatively 'normal' species, but at some point they acquired incredible powers to control matter and energy, space and even time itself.  Some Q maintain that they are omnipotent,  but at least one individual has claimed that there are in fact limits to their power  - although what these limits might be has never been ascertained. 
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The Federation's first contact with the Q came when one of them intercepted the Enterprise-D on its maiden voyage. Appearing in a variety of costumes, Q placed the ship's senior officers on trial for the 'crimes of Humanity' - although the crew convinced Q to release them,  years later he claimed that this trial was a never-ending process. 
In subsequent visits Q demonstrated a mixture of astounding power and knowledge combined with an almost childish sense of self-importance and selfishness.  On his third visit to the Enterprise, Q reacted negatively to an offhand comment by Captain Jean Luc Picard; he threw the ship across thousands of light years to the system J-25, leading to first contact with the Borg and eventually to two devastating attacks by that species.  He constantly berates and belittles other lifeforms and takes every opportunity to assert his own perfection. 
During her travels in the Delta Quadrant, the USS Voyager encountered a second Q - known as Quinn - who had been imprisoned in the nucleus of a comet for political dissension against the Q. Quinn was dissatisfied with a life in which there was nothing left to explore to accomplish, and had sought to end his existence. Q appeared on Voyager and a hearing was held which resulted in Quinn having his abilities removed so as to allow him to lead life as a Human. Despite this offer, Quinn took his own life shortly afterwards with the help of Q. 
This act apparently triggered a civil war within the Q continuum; Q proposed to end this conflict by mating with Captain Janeway of the USS Voyager in order to produce a new breed of Q - an offer the Captain forcefully declined. Later the crew of Voyager where taken to the Q continuum where, given the abilities and weapons of the Q, they where instrumental in ending the war. 
Q troubled Voyager on relatively few occasions, but he did place his young son in Janeway's care for a time in an effort to teach the boy some sense of responsibility. This was actually somewhat successful,  though it is hard to believe that a Q could ever really reform completely.