Sadao Maou
Back in Ente Isla, they were mortal enemies. Emilia's village had been destroyed by the Demon King's forces, and her father was presumed killed. In order to take him down, she took up her duty as the Hero and trained harder than ever, rallying forces against the dark legions and leading them into battle. Emilia admits later on that she had been obsessed with the idea of revenge, relishing the feeling of battle as she sliced off one of Satan's horns, and chasing him through the Gate when he fled Ente Isla. It is his defeat at the hands of the hero that forces the Demon King to retreat to Earth, resulting in the subsequent loss of his powers and much of his pride.
They did not recognize each other at first when they met again on Earth, she discerns his identity sometime later and openly confronts her nemesis in the streets, pulling a knife on him in an attempt to finish the tyrant. Strangely he does not fight back and instead chooses to run, the scrap ending in a draw when the police pull them aside and assume they were having a lover's quarrel. While she begins to dedicate her time to spying on him in case he reverts to his demon ways, Maou remains more irritated by her presence than anything else, and despite her long hours of surveillance, she can find no trespass in his actions. Growing tired of watching him, she eventually questions whether or not he even intends on returning home, though the Demon King claims he will subjugate both Ente Isla and the Earth one day, prompting her to stick around and keep an eye on him.
Though he still expresses a desire to return to Ente Isla and reclaim the lands there, he shows very little antagonism or ill will towards the Hero, and even begins showing her kindness and concern. This is first shown when they help each other escape an assassination attempt from an unknown assailant, and later when he orders Ashiya to bring the Hero some medicine and personally treats several of her wounds. Emi eventually confronts him over this change in personality, noting he is much different from the Demon King she fought back home. When mourning the losses she suffered when his army destroyed her home, Maou admits that back then he never truly understood humans, and genuinely apologizes, much to her confusion. Due to the loss of her father, Emi viewed Maou as a despicable person who thinks nothing of creating destruction and bloodshed. As such, seeing his kindness shocked her extremely, for she could not reconcile her idea of the Demon Lord with his human self.
"Family Photo"
While they still remained a little antagonistic to each other, the two eventually came to a truce. They have since come to understand each other better, and regularly join forces to combat the continued emergence of threats from Ente Isla that pose danger to their new home and friends. On occasion they both acknowledge that the day may come when both will be enemies again, though as the story continues, this becomes less apparent. Due to how often she interacts with Maou, Emi has grown mostly relaxed around him, much to her surprise. At time it is almost as though their enmity never existed. The dynamic of their relationship first improved when they began taking care of the sentient Yesod fragment, Alas=Ramus, who took the form of a young child that believes him to her "Father" and Emi the "Mother". Since then, the two have steadily grown closer while being forced to act as a family, with both slowly falling into the roles of parents over time.
After learning that her father is still alive, the Hero loses much of her previous bitterness towards the Demon King, but her emotions towards him becoming increasingly conflicted. Since her desire for vengeance had been the only thing driving her forward, she begins questioning her role as the Hero.
While Emi is held hostage in Ente Isla, she finally reconciles her past image of Maou as the evil Demon King with the reality of his kind and caring personality. Although she cannot fully forgive him because his invasion had tragic consequences for the people of Ente Isla, the anger that was driving her outward animosity towards him is no longer there. After being rescued by Maou and company, Emi's demeanor with Maou changes drastically. Emi's gentler disposition and more recent displays of affection are shown to confuse Maou, prompting him to react awkwardly due to his lack of familiarity with this side of her.
Emi having a "serious malfunction"
Maou becomes extremely concerned for Emi's mental health following her return from Ente Isla and the arrival of her long-lost mother who has secretly been pulling the strings from the shadows. As Emi experiences a mental breakdown resulting from this, she openly clings to Maou for support in spite of how her actions might be misinterpreted. Despite Maou's efforts to mask his attempts to help her under harsh words and actions, Emi is able to see through him, demonstrating that she has come to understand him more deeply.
While it is unclear whether Maou and Emi are harboring the beginnings of romantic feelings towards each other, it is worth noting that there have been hints of some level of attraction throughout the LNs. At their initial meeting in Japan when they did not know each other's identities, Emi's radiant smile made the dark feelings in his heart fade away and he felt awkward when he left the scene. This smile has a lasting impact on him, and he is reminded of it again during an event following Emi's rescue from Ente Isla. During the confrontation with Lucifer and Olba, Emi is shocked when she realizes that she found the smile on the newly restored Devil King's face to be charming/adorable. When Maou offered to show Emi a new world after naming her a Demon General, he found her flushed expression to be very cute and wanted to see more of it. After Emi shows him a smile after seeking comfort from him, he feels that her smile is precious but does not know the reason he feels that way.
Shirō Ashiya
During their first encounter, Emilia fought with the Demon General Alciel for three hours and nearly defeated him. He intended to use his death to take her out but was stopped by the Demon King who implored him to retreat instead. Emi and Ashiya share a cordial acquaintance on Earth, speaking to each other civilly despite a lingering sense of dislike between them. Due to her hatred of the Demon King's past exploits back home, Emi did not hesitate to put down Maou in front of Ashiya, which aggravated him to no end. In addition to this, she also found his frugal nature to be quite irritating, often pointing out his tightfisted habits, or exploiting them in order to coerce information from him.
By the the time Ashiya is kidnapped by Gabriel and brought back to Ente Isla, his ties with Emi are strong enough that he unconsciously starts devising an escape plan for the both of them. The two are forced to play along with Heaven's schemes and commence in a 10+ hour mock battle to stall for time. Emi apologizes for the demon casualties wrought in her name, but Ashiya reassures her instead of blaming her. Their relationship appears to have improved quite a bit after their return - she is far more polite to him and he even smiles at her during her gratitude "showdown" with Maou. Given that Emi is no longer being mean to Maou, and that was a major point of contention between Ashiya and Emi, their relationship may further improve with time.
Rika Suzuki
Her friend and colleague at Dokodemo, Rika appreciates Emi for not prying into her past when the former revealed that she was originally from Kobe. Rika is usually seen hanging out with Emi at work and in their free time. After Emi refused to join her for dinner after work, Rika became curious about her friend's personal life, eventually finding out about the hero's hostile relationship with Sadao Maou. She initially assumes the two are dating, much to Emi's horror, but her investigations into the topic eventually Rika to meet Ashiya, who clears away the misinterpretations with a story of his own. While highly supportive of her friends, Rika has a tendency to pry into Emi's personal business, though the latter takes it in stride, never holding it against her.
Chiho Sasaki
They did not get off to a good start, as on their first meeting Emi warned Chiho about Maou's character and unwittingly mislead Chiho into thinking that she had a relationship with Maou. Now, she is on friendly terms with Chiho, the only human on earth that knows the true identity of the people from Ente Isla. At times, she gets jealous of Chiho's chest size. Throughout the light novels Emi grows to become more protective of Chiho along with Maou. Their relationship gets a little shaky when Chiho starts becoming jealous of the fact that Emi and Maou are growing closer a fact that causes her to later breakdown. It is also shown that both Chiho and her mother think that Emi is a respectable and trustworthy person.
Gabriel
Earlier on in the light novel at least, Emilia is shown to be very hostile, foul-mouthed, and impatient to Gabriel lashing her sword at him with sometimes just as much as eye contact. Gabriel tried to capture her daughter Atlas-Ramus and like all angels except Lailah, Lucifer, and possible Sataniel, greatly angers her daughter. She considers him her enemy and she states that she shows no kindness to them especially for those that hurt her daughter.
Camio
Since Camio is a member of the Demon King's army, Emilia was initially very hostile to him. This is most notable shown when in his weaken state Atlas-Ramus tried to pluck his feathers, which really hurt him and so tried to go way only for her daughter to chase after him and accidentally trip. She blames her daughter tripping on him, and grabs him almost choking him threatening to kill and eat him as part of her curry only stopping when Chiho told her of the fallacy on her accusation. She earlier also threatened that she will kill all the demons that he tells about her holy sword scaring him in the process.