Like many of his moves, it is a parody of an existing "shotokan-style" move (in this case, the Shoryuken), and is likely the result of Dan's limited understanding of the Shoryuken, gained while he was briefly under Gouken's tutelage.
Executed the same as the Shoryuken, the Koryuken is also visually similar to the original move - a jumping uppercut with the fist half-extended, and the knee corresponding to the punching hand raised.
In the Street Fighter IV series, the move appears similar to a Tiger Uppercut, with Dan raising the knee opposite the punching hand. Unlike most Shoryuken-type attacks, Dan instead strikes with his leading hand in the said game, similar to the Forbidden Shoryuken. Curiously, in the Street Fighter IV series, Dan performs a proper Shoryuken for his time-over victory pose.
The punch button used determines the damage, distance, and height; light punch has the least height, damage, and distance, while hard punch has the most, and medium is a balance of the two.
As with the Shoryuken, this move functions well as an anti-air. In Street Fighter Alpha 3, Dan may occasionally flash before executing the move, gaining invincibility during the attack. Though the flash appears random (and thus lacking in tactical value), it actually appears to be determined by a set pattern. If the move is performed as the 23rd action since the start of a round, it will be a flashing Koryuken, as will every Koryuken perform as his 15th action from that point. While this technically gives the flash some tactical use, it is heavily limited.
In the Street Fighter IV series, the invincibility of the move is instead determined by the button(s) used, with light giving none and the new EX Special version, called the Saikyo-Koryuken, giving the most of the other three. The EX Special also hits twice, but deals less damage than the heavy punch version.