"The Arcana is the means by which all is revealed... Only courage in the face of doubt can lead one to the answer..."
—Nyx Avatar, Persona 3
In tarot readings, the Emperor symbolizes the desire to control one's surroundings, and its appearance could suggest that one is trying too hard to achieve this, possibly causing trouble for others; some elements in life are just not controllable.
Gameplay-wise, the Personas of the Emperor Arcana are Personas typically excelling in lightning-based and physical attacks; these Personas are almost always kings or emperor figures, but may simply be important male figures and deities. Typically, the king of a mythological pantheon is of the Emperor Arcana.
Within the Persona universe, characters of the Emperor Arcana are often male leaders or father figures (or both). Often times, the Emperor character is troubled by something very personal and doesn't know how to deal with it.
The social link for this Arcana can be started on April 27 when the Protagonist talks to Ms. Toriumi about joining the student council. Council meetings are held on Mondays, Wednesdays, and Fridays. Upon completing the Social Link, the protagonist gains access to the ultimate form of the Emperor Arcana, Odin, the Storm God.
The Emperor Arcana is also the Persona Arcana of Akihiko Sanada, the member of SEES.
The Emperor Arcana is represented by Kanji Tatsumi, who goes to the same school as the protagonist. Kanji is introduced partway through the game as a first-year student who often gets into fights and into trouble with the local law enforcement. As the protagonist attends the social link, Kanji reveals that he enjoys sewing and other crafts but has kept those interests hidden because of peer pressure. Kanji lost his father early in life and has since struggled to understand what it means to "be a man." Later in the social link, he meets a young boy whose whole family takes a liking to Kanji's hand-sewn dolls. This experience eventually gives him the confidence to accept his strengths and interests despite public opinion. Rather, instead of letting people say what they want, Kanji will work towards making sure he is properly understood.
Kanji's initial Persona is Take-Mikazuchi. At the maximum social link, Take-Mikazuchi transforms into its ultimate form, Rokuten Maoh, and bestows you the ultimate form of the Arcana, Odin, the God of Storms.
Leveling the Emperor Arcana up to Lv. 3 allows Kanji to learn his distinct follow-up move, Atomic Press. This attack strikes all enemies with a high percentage of dealing critical damage, which results in a Down status and the potential for an All-Out Attack. The follow-up move occurs at random and only after the protagonist has rendered a separate target Down first (and at least one other target is not currently Down).
In Persona 4 Golden, Kanji can unlock a third-tier persona: Takeji Zaiten. This is done by initiating Social Link meeting at any point after 1/10 if the Emperor Arcana is at Rank 10.
The Emperor Arcana is represented by the Confidant with Yusuke Kitagawa, which ranks up as the protagonist interacts with him. This Confidant unlocks Skill Card duplication and various team member skills for Yusuke as it ranks up.
As the Confidant reaches maximum Yusuke's Persona, Goemon, will transmogrify into Kamu Susano-o. Completing the Emperor Arcana Confidant will also bestow the ultimate form of the Arcana, Odin.
In Persona 5 Royal, if the Emperor Arcana confidant is maxed out and the protagonist chooses to reject the false reality during the events of the third semester, Yusuke Kitagawa will unlock his third tier Persona: Gorokichi, by fusing Goemon and Kamu Susano-O.