Its release date within the United States was July 2, 2003.
Warning: Wikipedia contains spoilers.
In the absence of the previously scheduled Judgment Day, John Connor has gone from messiah to burn-out and is wandering aimlessly around Southern California. Unfortunately it seems the apocalypse was only postponed by his actions in the 2nd movie, and the Air Force has taken over the Skynet project where Cyberdyne left off. A new pair of Terminators are sent back to alternately kill and protect Connor and his future wife and resistance lieutenants.
In this sequel, the T-850 Model 101 (played by Schwarzenegger) is again the protector, keeping Conner and Kate Brewster (played by Danes) safe from the T-X (played by Loken), an even more advanced Terminator sent from the future to terminate Conner and his future lieutenants, including Brewster.
In a plot twist, this particular model of T-850 reveals that he is in fact the murderer of John Connor in the distant future and that his wife (who will be Brewster) reprogrammed him for the mission. This plot point serves to illustrate the complete lack of both emotion and personal will that the machines possess. In the end, Brewster's father, an air force officer in charge of the Skynet project, tricks Connor and his daughter into taking shelter in a nuclear blast shelter by telling them that that is where the Skynet mainframe is located. The revelation that 'Skynet' is in fact a distributed computer program serves to update the film to modern technology which was not present at the films original conception.