Adria was born to Vala Mal Doran in Season 10, and was rapidly aged by the Ori. Vala gave her the name "Adria" telling her that it was her mother's name and that her mother was "a very strong woman", however, she later revealed to Daniel Jackson that the name was in fact her stepmother's and she was "a witch of a woman."
Essentially a genetically advanced human infused with Ori knowledge, she was created to circumvent the rules that the Ori cannot directly use their powers to conquer the Milky Way galaxy. She possesses several Prior-like superhuman abilities, and led the armies of the Ori until her Ascension.
As the Ancients prevent the Ori from taking direct action in the Milky Way, the Ori are forced to rely on their human followers to do their bidding in this galaxy. Similar to the Priors, Adria was created to circumvent the ascended "rulebook", and is essentially a genetically advanced human with Ori knowledge, a so-called Orici. It was her purpose to lead the armies of the Ori in the Milky Way galaxy, converting the galaxy to Origin in the process.
The fact that she is human is made evident by her lingering affections for her mother, Vala Mal Doran, despite being separated from her early after she was born.
According to Tomin's comment, "From one, all will learn", the Orici carries a tremendous amount of knowledge. In this respect, Adria is similar to the Harcesis, Shifu, the human child of two Goa'uld hosts, born with all the knowledge of the Goa'uld. However, because the Ori's knowledge of the universe is supposedly limitless, the Orici is not a true Ori. Yet, she is as close to the Ori as her human physiology allows.
Powers and Abilities
Adria was given several superhuman powers, similar to the powers of Priors. However, she is technically not a Prior, and is between the Doci and the Ori in status.  Her power is at least on the level of that of a Prior, although many of her abilities have yet to be revealed. Adria claims that despite the fact that she is essentially Ori, there is a part of her that is human, which somewhat limits her ascended powers.
She has been known to use telekinesis, and was called a "truth seer" by Tomin, suggesting she possessed telepathic powers. Adria also possesses the ability to heal life-threatening injuries in seconds. She can also use her telekinetic powers to dial a Stargate with her mind. She also possesses pyrokinesis. She was also granted a large amount of genetic knowledge, although this knowledge is limited by her human physiology. The knowledge in her mind unfolded as her body matured, causing headaches during her growth.  What exact knowledge she possesses remains unknown, but it does include at least basic information such as reading and writing, knowledge of Ori religion and the Ancients, and how to speak and read Ancient. She was also genetically engineered to rapidly mature after birth.
Although technically not a superhuman power, Adria is protected by a personal force field in the form of a pendant around her neck. It is powerful enough to protect her from basic machine gun fire, and the "fire breath" of a dragon and even the effects of the Dakara Superweapon. She claims it contains a part of the Ori city of Celestis, but to what extent this is true is unknown.
After Vala Mal Doran managed to prevent the construction of a Supergate at Kallana, she was transported to the Ori galaxy, where she was forced to live in the village of Ver Isca, posing as a loyal Ori follower. Apparently, she was chosen by the Ori to carry one of their own in human form, and was impregnated by them.
As the followers of Origin have strict rules against becoming pregnant out of wedlock, Vala was forced to marry Tomin, a fellow Ver Iscan villager. The timing eventually proved to be close enough and Tomin became convinced the child was his. However, both eventually learned from a Prior that Tomin was unable to conceive a child, forcing Vala to inform Tomin about the miraculous nature of the baby.
Vala carried the child for approximately nine months in Ver Isca, and eventually returned to the Milky Way aboard one of the Ori battlecruisers invading our galaxy at the Battle of P3Y-229. The Orici was born some hours after the battle, and was quickly separated from her mother.
Though Adria's gestation occurred at a normal human rate, she was genetically engineered to reach full maturity in a matter of days following birth. Within hours of being born Adria had reached the apparent age of four. However, she remained unnamed at the time. Instead, she was simply referred to as Orici. She was finally introduced to her mother at this apparent age as the Prior allowed her to visit her mother.
Adria used her abilities to stop the pain her mother was in after giving birth hours previously. She also remarked that her mother did not believe, and that she would eventually "find the path".
By the apparent age of seven, she was given the name Adria by Vala, after her own stepmother, whom Vala deeply despised. However, Vala claimed it was the name of her real mother, describing her as a "strong woman". Adria also tried to convert her mother to Origin, and claimed that it was the Ancients who tried to destroy the Ori.
When Adria had reached the apparent age of twelve, the Ori fleet had reached the planet of Chulak and had begun a ground incursion. In the mean time, Vala had discovered Daniel Jackson, who had ringed aboard during the battle, and the two subsequently attempted to capture Adria and get her to renounce the Ori.
Their plan backfired, and Daniel was captured by Tomin. Adria ordered Jackson to be killed, but as Tomin fired his weapon, Vala moved in front of Daniel and was hit by the blast. Adria then began healing Vala, and while she and Tomin were distracted, Daniel was able to stun them both. The two subsequently attempted to leave the room, but a Prior blocked their exit. However, the Odyssey arrived at the time, and attempted to beam Daniel off the Ori ship. As he was beamed away, Jackson grabbed Vala, and the two were both transported away safely.
By her adult age, Adria had taken a primary role in leading the Ori forces in spreading Origin. According to her estimates, it would take the Ori less than a year to conquer the entire galaxy.
On an unspecified planet, Adria spoke to the converted inhabitants but was attacked by the Dakara Superweapon now modified to destroy all living matter as a desperate strike against the Ori. Despite the eradication of the planet's entire populace, including a Prior, Adria had survived the attack thanks to the personal shield embedded in her necklace. She then used her telekinetic abilities to kill the Jaffa who had boarded the vessel, and forced them to reveal the location of the weapon. Adria then disabled explosives which SG-1 had placed near the engine core and single-handedly piloted the Ori vessel to Dakara where she annihilated the Superweapon and conquered the planet. During this appearance, she revealed that the Ori have 'plans' for Daniel, which may tie in to Daniel apparently becoming a Prior in the episode "The Shroud".
Some weeks later, after SG-1 had discovered the supposed location of the Sangraal and had travelled to the planet in question to search for the weapon, Adria posed as a frail librarian named Osric. She then attempted to trick SG-1 into obtaining the Sangraal for her. While she knew the location of the Sangraal, only one of "pure heart" could obtain it. Since Daniel Jackson was once ascended, she believed that he would be able to obtain the weapon for her.
Adria helped SG-1 to find the location of the weapon by providing Vala Mal Doran with the necessary information via a dream. Due to safeguards put in place by Morgan le Fay, Adria could not use her abilities. When the team visited the planet, she forced them to aid her in locating the weapon, bent on later stealing it from them. After SG-1, assisted by both Adria and Ba'al, were successful in overcoming the many trials set up by Morgan le Fay, all but Adria were transported away. As the locals had refused to accept Origin, Adria burned down the village before attempting to relocate SG-1.
However, her attempts at finding their whereabouts were problematic at best. In an attempt to hide the Sangraal, Morgan le Fay had connected the transporter system to a sophisticated program that would periodically transport the ones who were originally brought to the planet to a new random world outside the normal Stargate network.
In the meantime, SG-1 discovered that Merlin had never completed the device. Instead, he had been placed in stasis by his old rival, Morgan le Fay, so that he could one day construct the weapon. When he was revived a thousand years later by SG-1, he began work on the device, but soon realised he would not live long enough to complete it. Instead, he reconfigured a Repository of Knowledge to download the information necessary for constructing the Sangraal into Daniel Jackson's mind. Still, Jackson also failed to complete the device in time, and was captured by Adria, who had eventually been able to locate them, before his work was done. 
Adria attempted to convert Daniel Jackson to the path of Origin in the hope that this would allow her to convert both Earth and her mother. She believed she had succeeded and transformed Dr. Jackson into a Prior and had him complete the Sangraal on the idea that it would be used against the Ascended Ancients. She stopped Dr. Jackson from completing the weapon and sent him to convert planets to Origin in order to build up the Ori's strength for the final battle. However Jackson betrayed her and returned with SG-1 who had commandeered her Ori Battlecruiser. Their plan was to deactivate the Supergate, which was blocked by a connection to a small regular Stargate in the Pegasus Galaxy, and reconnect it to the Ori home galaxy in order to use the weapon. She was tricked once again by Daniel Jackson who feigned loyalty to her before using his enhanced Prior powers to render her unconscious. She was then last seen on her battlecruiser which was sent through the Supergate with an activated Sangraal.
In the Stargate SG-1 episode "Dominion", Adria located Vala, and planned to search with her for the infinite treasure known as the Clava Thessara Infinitas. However, it was all a trick set by SG-1 to capture Adria and force her and the Ori armies to leave the galaxy. But Ba'al intervened and captured her instead, subsequently implanting her with a Goa'uld symbiote; a clone of Ba'al's. SG-1 then re-captured Adria from Ba'al's ship and, with the help of a Tok'ra doctor, was able to remove Ba'al with the intention of implanting a Tok'ra symbiote to suppress Adria's mind and tell SG-1 of her knowledge of the Ori's fate after SG-1's attack in "The Shroud". However, as Ba'al was removed from Adria's body, he released a lethal dose of a nerve toxin into her bloodstream. As she lay dying, her captors decided to administer a more lethal dose of the toxin, as her body is too weak to take on the Tok'ra symbiote. Just as the poison is about to be injected into her bloodstream, Adria awakens and throws the Tok'ra doctor aside with her mind powers. Moments later, with her mother watching helplessly and the rest of SG-1 unable to reach the room in time, Adria ascends to become one of the Ori - now a greater threat to the galaxy than ever before. However, this also means that she can no longer interfere in the lower planes in the Milky Way without inciting the Ancients, leaving the Ori army effectively leaderless. Either way, SG-1 feels that they will see another incarnation of her along the way.