The triangle offense is most commonly associated with Phil Jackson, who used it when he coached Michael Jordan and the Chicago Bulls and when he coached Kobe Bryant and the Los Angeles Lakers. He did not invent the offense, but he is the coach most connected to the offense in the NBA.
The triangle offense is one that relies much less on set plays than other offenses and more on "reading" and reacting by the players on the court. It is based on the idea of constant movement and passing, which is why coaches trying to use it in the NBA often come into conflict with stars who would rather hold the ball more of the time.
The basic idea of the triangle offense is that there should be one center and four other players who are more or less interchangeable. The center must be able to pass because much of the offense is predicated on passes into the post and movement around it. It is known as the triangle offense because its most common set is one in which there are three players forming a triangle on the ball side of the court with two others on the weak side to create spacing and to make cuts and receive passes.
Needless to say, there is much more to the offense than this. The offense is considered one of the hardest to learn due to the fact that it demands understanding of the game more than the ability to simply run designed plays.