Red Raven is the mental embodiment of Raven's inner rage, hatred, anger, pure dark, and evil essence inside of her, the inter-dimensional demonic heritage from her demon father Trigon. Rage, unlike the other Emoticlones, possesses a far darker voice and four gleaming bright red glowing eyes (Though in the comics, she had two for a brief moment).
Out of all of Raven's Emoticlones, Rage is the most dangerous; because of this, Raven must keep her constantly suppressed through intense meditation, for even the slightest expression of this emotion (such as in her battle with Doctor Light in "Nevermore" and Terra in the two-part second season finale "Aftershock") can cause her to go out of control and become much more demonic like her evil demonic father, Trigon. When this dark form is manifested, Raven's supernatural, mystical abilities and powers are at their strongest and most powerful.
During the events of "Nevermore", Rage had escaped from Raven's control and was causing great mayhem throughout her mind in the form of Trigon, but thanks to some advice from Beast Boy and Cyborg, Raven fused with her other Emoticlones and went into White Raven form, causing her to once again gain control of Rage.
Rage later returned in the Teen Titans Go! comic book issue Red Raven (a continuation of "Pieces of Me"), where she escaped into the world and began destroying anything and everything that seemed related to the term "father", such as the Founding Fathers exhibit at the museum and Father's Day items in stores. This indicates that Raven hates Rage so much, even after she banished her, that she wants to cut off all ties with her for good.