The Department of Extranormal Operations, often shortened to just D.E.O., or DEO, is a fictional government oversight agency featured in comic books published by DC Comics. They are part of the mainstream DC Universe that existed following Crisis on Infinite Earths and before the "New 52" continuity reboot. They first appeared in Batman, Volume 1 #550 in January, 1998. The D.E.O. was created by writer Doug Moench. The organization served as the main backdrop to the Chase comic book series, which was published from February-November, 1998. The D.E.O. also served as the main focus of the 2015 CBS television series Supergirl.
The Department of Extranormal Operations is a top-secret U.S. government agency, whose purview is to monitor and control all alien activity in the United States. This includes detainment of alien beings that the D.E.O. feels is a threat. As their existence is known only to a few, the D.E.O. has been known to operate outside of America's borders. The agency functions under the auspices of the United States military. Its central headquarters is an underground bunker located just outside of National City. The D.E.O. was likely founded shortly after the first public appearance of the alien hero known as Superman.
The director of the Department of Extranormal Operations was a man named Hank Henshaw. Henshaw was a militant reactionary who saw all alien beings as a threat. He fostered a "shoot first, ask questions later" paradigm while working in the field. One of Henshaw's top agents was a man named Jeremiah Danvers. Danvers had a unique outlook on extraterrestrial phenomena as his own adoptive daughter, Kara, was in fact an alien.
Around the year 2005, Henshaw and Danvers responded to a reporter about an alien sighting in the jungles of Peru. Danvers was the first to encounter the being, who turned out to be a refugee Martian named J'onn J'onzz. The Martian saved Danvers' life when he was nearly attacked by a snake. Jeremiah realized that this being was a good man, and not deserving of the type of treatment typically fostered upon such an individual by the D.E.O. As it turned out however, Henshaw came upon the seen and immediately opened fire on the Martian. Danvers tried to stop him, and in the scuffle, suffered a fatal stab wound to the abdomen. Henshaw perished when he fell from the top of a cliff. As Jeremiah lay dying, he asked J'onn J'onzz to watch over his daughters. This was a promise that J'onn sought to uphold the remainder of his life. 
J'onn used his innate Martian abilities of shape-shifting to assume Hank Henshaw's form, as well as his position as director of the DEO. As years passed, he found Jeremiah's first daughter, Alex Danvers, who had hit a low point in her life, and was arrested for public intoxication. He approached her directly and offered her a position with the D.E.O., knowing that she had the discipline and moral fiber to become an accomplished agent. J'onn was proven right, and Alex went to train with the D.E.O., ultimately becoming his most valued and trusted agent. At this time, Alex had no idea of J'onn's true origins and motivations. 
It wasn't long before the D.E.O. found that they really had their work cut out for them. The same spatial anomaly that first brought Kara Danvers to Earth, also brought with it alien criminals and refugees that had been imprisoned in Fort Rozz in an interdimensional plane of existence called the Phantom Zone. Many of these criminals were Kryptonian, just like Kara, but others, such as Vartox, hailed from other worlds. In addition to hunting down and capturing as many of the Fort Rozz refugees that they could find, the D.E.O. also apprehended other alien menaces such as the telepathic being known as Jemm.
When Kara Danvers was 24-years-old, she reached the peak of her own Kryptonian abilities and decided to follow the path of her illustrious cousin, Superman, by bearing his symbol and taking on the costumed identity of Supergirl. She quickly became the resident hero of National City. Since many of Supergirl's foes were also wanted by the D.E.O., it made good sense that Kara would form an alignment with them. Through her sister Alex's influence, Kara became an unofficial operative of the group, and J'onzz was able to continue to keep his promise to Jeremiah by keeping a glowing red eye on both daughters.
Alex Danvers was never satisfied with the official reports concerning her father's disappearance, and so she began investigating the matter on her own privately. Her inquiries began shining a light on Hank Henshaw, and she began to suspect him of wrongdoing. Alex finally learned the truth during an incident wherein Jemm escaped from prison, and Hank was forced to reveal his true identity to Alex. Although he had been leading a double life, he convinced her that he was a friend to her father, and had always watched out for the Danvers family. Soon after, Kara learned the truth about J'onn J'onzz as well. 
The D.E.O. and the Danvers sisters crossed swords with numerous villains including Kara's terrorist aunt, Astra, the electrically charged Livewire, the killer robot Red Tornado, a Bizarro version of Supergirl, an alien computer construct known as Indigo, and even billionaire industrialist Maxwell Lord.
During a time when Kara had been corrupted by the influence of Red kryptonite, J'onn J'onzz was forced to reveal himself as an alien in public. He knew that this would mean the end of his time with the D.E.O., and possibly even his life. Not wishing to make matters worse, he voluntarily surrendered himself to the authorities. 
The United States government sent in U.S. Army General Jim Harper to interrogate J'onzz. Harper had been close friends with the real Hank Henshaw, and was determined to see justice done, not matter the cost. After the interrogation, Harper decided that J'onzz was going to be transferred to Projet Cadmus where he would be dissected. With some help from Alex, J'onzz escaped confinement and used his telepathic abilities to make Harper forget the entire affair. Afterward, he named Lucy Lane as the new director of the DEO. 
During Lane's first day on the job, she officiated the observation of a young woman named Siobhan Smythe, who had been brought to the D.E.O. for testing after demonstrating the power of sonic projection. After being studied by Doctor Amelia Hamilton, it was determined that Siobhan was not an alien or a metahuman, but something else entirely. Siobhan soon became the super-villain Silver Banshee, and used her powers to free the D.E.O. detainee, Livewire. 
- Maxwell Lord is the first human being to become a detainee of the D.E.O.
- The D.E.O. has some connection to another government alien think tank called Project Cadmus. While the D.E.O.'s functions are to detain alien refugees, Project Cadmus's objectives involve scientific research, and even dissection of captured alien samples. Their existence is why Superman does not do government work.