Kurama

The is the most powerful of the Tailed Beasts; a single swipe of one of its nine tails can raise tsunamis and flatten mountains. After the demon fox attacked Konohagakure, Minato Namikaze, the Fourth Hokage, sealed it within his newborn son, Naruto Uzumaki, by using the Dead Demon Consuming Seal, sacrificing his own life to do so.

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Personality
The demon fox is a giant beast of almost complete malevolence. It is intelligent, and has a sadistic and sarcastic personality. However, it does have a sense of honor, and possesses a loathing respect for Naruto and Minato. Additionally, it has a distinct sense of pride and became rather angry and scornful at Naruto when he rejected its chakra in Part II, though this might have merely been anger over missing an opportunity to partially usurp control over Naruto's mind and body and thus weakening the seal. The demon fox relishes the thought of killing Naruto, but it knows it has to help as long it's sealed within him or it will die.

The demon fox will go to great lengths to preserve its own life, as shown when it forces its chakra into Naruto whenever his life is in danger. However, it will back down if Naruto threatens to harm or kill himself.

Sasuke and Sai Arc
For the most part of Part II, the demon fox stayed quiet, mainly just giving Naruto power to enter Tailed forms throught the Rescue Gaara arc. As Naruto battled Orochimaru, Naruto slipped through the bars of the seal and the demon fox concentrated some of its power into Naruto to make Naruto going into a Four-Tailed Form where the demon fox temporarily gained limited control of Naruto's body. With this, the demon fox made Naruto's body wreck havoc on not only Orochimaru but on every living thing around it, including Naruto's ally Sakura. Tenzo, codenamed "Yamato", stabilzed the transformation and forced the Demon Fox back after this.

Later, as Naruto and company ran into Sasuke again, the demon fox urged Naruto to show Sasuke "their" power. It tempted Naruto to remove the seal so their power could be one. As Naruto refused, the demon fox grew angry and mocked Naruto that he couldn't do anything by without it. However, Sasuke interfered and the demon fox commented that Sasuke's chakra was just like the one person who had more cursed chakra than its own: Madara Uchiha. Sasuke then surpressed the demon fox and Naruto shunned it because of how it was hurting his friends. Needless to say, the demon fox wasn't too happy about this.

Elemental Training
As Naruto trained with Kakashi Hatake in wind element training, the demon fox started to "experiment" with Naruto's shadow clones by sending chakra into a few of them. However, Tenzo surpressed them.

Fire Temple Arc
As Naruto battled Sora (who was using the demon fox's chakra), the demon fox started to force Naruto to accept his power. It appeared it was angry that someone other than its Jinchūriki was using its power. However, the demon fox was stopped as Naruto began stabbing his leg and threatening to back the beast off. It begrudgingly pulled back but not before absorbing Sora's demon fox power.

Naruto vs Pain
The demon fox remained inactive for a long time after Naruto shunned it, it did however refuse Fukasaku when he tried to fuse with Naruto. But as Naruto battled Pain in the mostly destroyed Konoha, Naruto was pinned down by Pain and was about to be captured by the Akatsuki Leader. However, Hinata Hyuga arrived and declared her love for Naruto and as Pain nearly killed her, Naruto slipped into a six-tailed form and allowed the demon fox to take control once more. While in minor control, the Demon Fox took this time to destory the First Hokage's Necklace which was the key for surpressing its power. When Naruto is still overwhelmed by Pain, the demon fox convinces Naruto to take even more of his power, transforming into his 8 tailed form. Naruto attempts to break the seal completely and transform into his 9 tailed form, but the soul of Minato Namikaze appears and prevents Naruto from doing so. It is revealed that if Naruto ever goes into his 8 tailed form, Minato would appear inside his consciousness. One final safeguard to keep the demon fox under control.

Escape Plan
The demon fox, as expected, seems to have a plan for escape. When Naruto first met Sasuke after his defection, the demon fox told Naruto to remove the seal internally so that Naruto would have full access to its chakra. It has also been stated that if the seal is broken, or if Naruto draws too much of it's power, the demon fox would be released. As such, during Part II, the demon fox eagerly offers its power to Naruto when fighting against Orochimaru and during his later encounter with Sasuke. This plan, however, failed as Naruto refused to use the demon fox's power.

During Naruto's element training, the demon fox was forcing its chakra into one of Naruto's shadow clones. Its reason for doing this is unknown, but it apparently has something to do with its plans for escape.

During Naruto's battle with Pain, Hinata was greatly injured after trying to protect Naruto, causing him to slip into his six-tailed transformation. During this transformation, the demon fox destroyed the First Hokage's necklace as it tried to supress the demon fox's chakra. Since Yamato had stated that his suppression technique was dependant on the First Hokage's necklace being on Naruto, it's concievable that there was more to the crystal's destruction than the events of the time of destruction.

Later, when trapped by Chibaku Tensei, he advises Naruto to destroy all those who cause him pain, unleashing the Eight-Tailed Transfomation. Achieving mental dominance over Naruto he commands Naruto to remove the seal. But this attempt is ultimately foiled at the last moment by the spirit of Minato Namikaze, The Fourth Hokage.

According to the Third Databook, Nine Tails plans to take over Naruto's body, hence why he keeps telling Naruto to remove the seal.

Activity inside Naruto
As the demon fox's host, Naruto has accelerated healing, unrivaled stamina and speed, massive strength, and access to the demon fox's near-limitless chakra. The demon fox's chakra allows Naruto to perform his signature jutsu, Shadow Clone Technique, seemingly without limit. He is even able to summon Gamabunta, a normally Kage-level feat requiring a huge amount of chakra proportionate to the size of the summoned creature.

Naruto typically accesses this prodigious supply of chakra from the demon fox during extreme emotional states or life-threatening situations. When he first accessed it, the demon fox's innate rage took over Naruto's mind, causing him to strike out at whatever was near him at the time. Afterwards, Naruto learned to exercise more control over the chakra, and he can now retain his mental state when drawing upon small amounts of the demon fox's chakra. It is said earlier in Part I that Naruto's chakra had to overlap the demon fox's while still leaving some of his own chakra for himself. Kakashi has a theory that implies that Naruto may have slightly more chakra than the demon fox's. After Jiraiya's training, Naruto learns to summon a small degree of the demon fox's power at will. To do this, Naruto travels into his own subconscious (depicted as a sewer) and demands the demon fox to give him its chakra, which he calls "rent". The demon fox simply complies with him because it is amused by Naruto's bravery, and it knows that it will die if Naruto dies.

Hiruzen Sarutobi and Kakashi Hatake allude that if Naruto loses all control of the demon fox's chakra, the seal will break and the demon fox will be released. Though the seal is unlikely to break on its own, the chances of it happening increase as Naruto draws upon the demon fox's chakra. After Naruto trained with Jiraiya, he inadvertently released Naruto's four-tailed transformation. Jiraiya mentions that the seal may be becoming weaker over time. This is evident in Naruto's transformations, which occur much quicker and with greater intensity than they did in the first part of the series, often against his own will. As seen during Naruto's training sessions, even the slightest amount of frustration on Naruto's part allows the demon fox's chakra to leak out in excess. Naruto will then rapidly progresses to his four-tailed state.

Apparently, while within Naruto, the Mangekyo Sharingan can't be used to control the demon fox. However, someone with good control of a normal Sharingan is apparently still able to suppress its chakra (as demonstrated by Sasuke). Also, it rejects Fukasaku when he tries to merge with Naruto during his Senjutsu training for reasons unknown (although it can be safely assumed that it did that so Naruto would have no choice, but to only rely on its chakra).

When Naruto was captured by Pain's Chibaku Tensei technique, the fox first convinces Naruto to destroy everything that causes him pain and give his heart completely to the demon fox, so that the fox could destroy everything that was causing Naruto pain, thus distorting the seal to the point where the Eight-Tailed Transformation is released. He then goes on to gain mental dominace over Naruto and nearly succeeds in having the seal removed completely, but ultimately fails due to Minato Namikaze appearing in Naruto's mind to stop him.

The Seal
Jiraiya later reveals that Naruto's four-tailed transformation was his doing by using Gerotora, the toadkey with the same seal on his belly as on Naruto's. He had the toadkey "twist" the seal a little, which resulted in the demon fox's chakra taking over. Jiraiya mentions that Minato split the demon fox's chakra in two halves, sealing the Yin chakra with the Dead Demon Consuming Seal, thus making it inaccessible to Naruto, but sealing the Yang chakra with the Eight Trigrams Sealing Style, allowing Naruto to access it.

The seal also has a final safeguard to prevent the nine tails from escaping. Minato Namikaze made it so that if Naruto ever transforms into his eight-tailed form, Minato will appear in his consciousness in order to prevent the seal from being broken.

Trivia

 * Many incorrectly believe the demon fox's name is "Kyuubi", the direct English pronunciation of "Kyūbi". When the demon fox is referred to, it is usually called "Kyūbi", which stands for "Nine-Tails". This is similar to the belief that Minato Namikaze's name, before being revealed, was "Yondaime", which means "Fourth".
 * The demon fox's eyes were like that of the Sharingan when it was being controlled by Madara.
 * According to the 3rd Databook, Nine Tails is the only one who knows the Truth. Truth about the Uchiha clan and about Naruto. However, it appears that the Demon Fox has no intentions of telling what it knows