The Rasengan is a powerful technique invented by Minato Namikaze, though after three years of development it was never completed. Rasengan was meant to be an example of nature manipulation as Minato meant to combine his chakra element with the Rasengan. He died before he could accomplish this, but the Rasengan is still a very useful and difficult jutsu to learn.
In Part I, Jiraiya taught Naruto how to use the Rasengan, breaking the learning process into three steps:
The first step was learning how to burst a water balloon by spinning the water inside in multiple directions at once using only the user's chakra, which emphasizes rotation.
The second step was for the user 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 and final step was to combine both step one and two by spinning chakra inside a balloon without moving the balloon itself, emphasizing stability of the sphere shape.
Although most users learn to form the Rasengan with one hand, Naruto and Konohamaru use one or more shadow clones to help in its creation. By having the hands of the shadow clones act as shells, Naruto and Konohamaru can use the complete technique without having to dedicate extra time to mastering their chakra control.
Naruto has been seen using a one-handed Rasengan when in a One-Tailed Transformation.
The Rasengan doesn't require any hand seals to use, relying only on the chakra control of the user. Once it's complete it becomes self-sustaining, which does not give Rasengan a definite limit of use like its counterpart, the Chidori. The compacted nature and moving speed of the chakra allows the Rasengan to grind into whatever it comes in contact with, causing major damage. This also tends to launch the target back once hit. In an anime filler arc, Naruto was able to perform this jutsu as a mid-range projectile technique by somehow throwing it into a wall.
The Rasengan was designed to use the caster's elemental affinities. Minato was unable to add his affinities to his Rasengan before he died. Kakashi tried to add his lightning element to the Rasengan but was unable to and explains that he uses the Chidori and Lightning Blade instead, which requires a certain point of shape manipulation and nature manipulation itself. Naruto was the first to add his Wind affinity to his Rasengan and even advanced it to the point that he was able to create the Wind Release: Rasenshuriken.
While in training, Naruto improved the Rasengan with the Great Ball Rasengan, after Jiraiya pointed out during their time away from Konohagakure that it would be easier to hit an opponent with a larger Rasengan. Later, Naruto mixed his Wind natured chakra to start developing the Wind Release: Rasengan as it was originally intended. Naruto took the Rasengan even further than his father by creating the Wind Release: Rasenshuriken, a variant of the Wind Release: Rasengan. However, this new technique also damaged the user's arm and chakra circulatory system, making it a double-edged sword. Even so, Naruto overcame this obstacle as shown during his fight with Pain when, in Sage Mode, he threw the Rasenshuriken, preventing damage done to his own body.
Naruto can also wield a Rasengan in both of his hands while in Sage Mode. Sage Art: Great Ball Rasengan uses two Great Ball Rasengan, while Sage Technique: Rasenrengan uses two regular Rasengan. Jiraiya is suggested to be capable of doing something similar; when teaching Naruto the Rasengan, he does the balloon exercise in both of his hands.