The Q are often known as an "omnipotent" and immortal species with a great deal of control over space, matter, and time. The Q have been known to travel through time, create alternate timelines, move asteroid belts and stars, change universal states of nature, and teleport with great ease. They have also been able to transfer their powers to other life forms, and have their powers removed by a higher authority in the Q Continuum.
There are, however, concerns about the Q's omnipotence. In the Voyager episode "Death Wish", Q (who later named himself Quinn) informs the crew that the Q are not, in fact, omnipotent:
"You mustn't think of us as omnipotent, no matter what The Continuum would like you to believe. You and your ship seem incredibly powerful to lifeforms without your technological expertise! It's no different with us; we may appear omnipotent to you, but believe me, we're not!"
|The Q later known as Quinn - Deathwish|
There are also concerns about the initial state of the Q - is has been suggested (both by Q) that they had evolved over countless centuries into the "ultimate form of evolution", and that they had always been as they were, without a beginning.
|Q judges - Q2|
|Mating Q - Q and the Grey|
Quite accidentally, I have only included pictures from Voyager (the Q appeared in three VOY episodes), and most of the information gathered for this post is from Voyager as well - this is merely because the Q race was explored more deeply in this series. Q first appeared in the TNG pilot, "Encounter at Farpoint", and the Q appeared many times later - in 8 TNG episodes, and one DS9 episode.
Fun fact: I had always thought that Trelane (TOS The Squire of Gothos) might be a Q - a fact backed up by a book I recently read - Q-squared by Peter David.