The Rasengan is a technique invented by the Fourth Hokage, after three years of development, though it was never completed. The Rasengan was meant to be an example of nature manipulation as the Fourth ment to combine his chakra element with the Rasengan. Despite this, the Rasengan is still a very useful, if difficult to learn jutsu. As Jiraiya trained Naruto how to use it, learning the Rasengan was broken into three steps. The first was to burst a water balloon by spinning the water inside in multiple directions at once using only the users chakra, which emphasizes rotation, then one has to burst a rubber ball and since rubber is thicker and air is completely different from water, it forces the user to use denser chakra to break it, this step emphasizes power, the last is to combine steps one and two by spinning chakra inside a balloon without making it move at all.
Because the Rasengan doesn't use handsigns, it relies only on the chakra control of the user, and once its complete, it becomes self-sustaining. The chakra spinning at such a speed and compacted into such a small state allows the spinning chakra to grind whatever it comes in contact with, causing major damage to it, and can even cause internal damage. And because the Rasengan is just pure chakra, it doesn't have a definite limit of use like its counterpart, the Chidori has.
While the Rasengan is a one-handed technique, Naruto has to use three or more hands to use it, mostly due to his poor chakra control. He requires to use of his one hand to create the shell, and a clones hands to spin the chakra. When he goes into his one-tailed transformation, he was able to make a Rasengan using his hand to spin and the demon fox's chakra shield to create the shell. Because he mixed it with the demon fox's chakra, the Rasengan had a purplish glow, making it equal in power to Sasuke's Flapping Chidori.
While in training, Naruto improved the Rasengan with the Odama Rasengan, or Great Ball Rasengan. Later Naruto mixed his Wind natured chakra to create the Wind Release: Rasengan as it was originally intended.
The Great Ball Rasengan is a technique created by Naruto while training in the time skip with Jiraiya, prior to the release of the Shippuuden episodes. Although this technique requires Naruto to use a Shadow Clone, the result is a larger ball of condensed chakra.
An original jutsu by Naruto, expanding on the original concept for the Rasengan by the Fourth's ideals: mixing ones chakra element with the Rasengan. Naruto mixed his wind element, maintaining its original stablity and increasing its power, far beyond that of his Odama Rasengan. To create it, Naruto needs the aid of two clones, and in total, five hands to create, his hand to create the shell, one clone to spin the normal chakra, and the other to mix the wind chakra.
The Wind Release: Rasenshuriken is a variation of the Wind Release: Rasengan, similar to the Great Ball Rasengan being a variation of the standard Rasengan. By manipulating the wind chakra of the Wind Release: Rasengan, Naruto was able to create four large points, making the Rasengan appear as a giant shuriken, the Rasengan in the center remaining a perfect sphere. Despite the name, because the Rasengan is a handheld jutsu, the Rasenshuriken can not be thrown, instead Naruto has to figure out a way to get close enough to his opponent and hit them. If successful, the victim becomes trapped in a vortex of wind chakra blades that attack the body on a cellular level. The wind chakra severs the chakra and nerve channels in the body.
The main drawback of the Rasenshuriken is that it is a double-edged sword. While it can utterly reap any opponent trapped in the vortex, it also damages the users body. When Naruto used the Rasenshuriken twice on Kakuzu, it resulted him with limited cellular damage and a few broken bones. The Fifth Hokage has hence decreed the Rasenshuriken as a Kinjutsu, forbidding it as it could cause Naruto unable to mold chakra at all.
In his one tailed form, Naruto uses the demon fox's chakra sheild to spin the chakra while using his own hand as a shell to form a Rasengan. The appearence of the Demon Fox Rasengan In the anime, the Rasengan took a more purplish color rather than the standard blue, while in the second movie and the video games the Rasengan took a more redish color, similar to the demon fox's chakra.
Three-Tails Rasengan (三尾螺旋丸 Sanbi Rasengan)
While in his Three-Tailed Fox form, Naruto creates a clone that charges a Rasengan that implodes in on the enemy attacked by it.
Used only in the video games, the Great Flaming Rasengan is simply a standard Rasengan with added fire power from a Fire Release jutsu. It also has the same grinding effect as the Rasengan but with flames connected to it.
Rasengan of Truth
An advanced and longer lasting form of the Rasengan.
Special Attack Rasengan
Naruto punches the enemy to the ground and creates two clones as one forms a weaker Rasengan in the other's hand that the clone strikes the enemy with another resengan.
Shadow Clone Rasengan (連環螺旋丸 Renkan Rasengan)
Naruto creates a clone that charges chakra in Naruto's palm while kicking the enemy before disappearing, leaving the real Naruto to use the fully charged Rasengan.
Naruto kicks the enemy while creating several airborn clones that kicks the enemy to the ground as Naruto jumps to a clone that charges a Rasengan that Naruto uses on the enemy while falling.