The Phoenix is an alternate personality of Jean Grey that was manifested when her mutant powers were activated. The Phoenix persona is extremely powerful and violent. Due to its nature, it was sealed away in Jean's subconscious mind by Professor Charles Xavier.
When Jean Grey saw her best friend Annie Richardson killed in an accident, she underwent an emotional trauma that activated her powers. The combination of the loss of her friend and the emergence of her powers severely affected her, so much that an alternate personality emerged; one with an even greater control over her powers than Jean. This personality called itself the Phoenix. The Phoenix was utterly enraged at the loss of Annie, and it could have torn Jean's home and family apart. Charles Xavier saw the only way to contain it is by placing powerful mental blocks around the Phoenix personality, locking it away in Jean's subconscious mind.
The Phoenix persona later briefly reemerged once the mental blocks were weakened which is further compromised by Jean being unable to control her powers. Jean quickly isolated herself and her persona in the Danger Room from out of harm ways; with assistance from Superman and Scott Summers, Professor Xavier was able to rebuild the mental blocks with Jean's concentration that sealed the Phoenix away again.
When faced with the threat of the Friends of Humanity exposing mutants to the public and preparing to mount a mass assault with Sentinels, Jean retreated into her mind to make contact with the Phoenix once again when she required the Phoenix's power to find the currently-captured Professor. Aided by an aspect of Annie that had remained in her subconscious, Jean and Phoenix were able to reconcile, allowing Jean to draw on Phoenix's powers when needed.
In Marvel Comics, the Phoenix is known as the Phoenix Force, which is an immortal and mutable manifestation of the prime universal force of life and passion. Born of the void between states of being, the Phoenix Force is a child of the universe. It is the nexus of all psionic energy which does, has, and ever will exist in all realities of the omniverse, the Guardian of Creation, and of the dangerously powerful M'Kraan Crystal. The Phoenix Force has bonded with other characters, who often used the alias Phoenix. The Phoenix Force is famous for its central role in one of the most influential storylines in American superhero comics, The Dark Phoenix Saga, and is frequently linked to Jean Grey.
The Phoenix of The Last Son series bears more resemblance to the Phoenix of the X-Men movie franchise, where it is merely an alternate personality for Jean created when Xavier placed psychic blocks on her powers until she could control their more destructive potential.