Exceed (エクシード Ekushīdo) are a race of cat-like beings that exist in Edolas as far back as 100 years prior to the main storyline. Mystogan's Reverse Anima sucked the entire race to Earth Land.
The Exceed used to see themselves as divine rulers of the humans, guiding them according to the will of their queen, before they found out how weak they really were. The humans of Edolas generally hold them in high regard, as seen when Erza Knightwalker and a group of soldiers bowed to Happy and Carla upon their return home (not knowing at that time that the two were allied with the Earth Land Mages). Presumably, this is due to the fact that they are the only beings in Edolas whose Magic is located inside of their bodies, just like the Mages of Earth Land.
Due to this reverence, the Exceed see themselves as angelic beings and above humans in general Their ruler is considered as the God of Edolas. However, the only spell they can perform is Aera and, as Byro said, the limited Magic of the Exceed cannot compare with that of Earth Land's Mages. This created strife when they learned that Lucy Heartfilia was in Edolas and could still use Magic, thus threatening the Exceed's power. As a result, their queen gave direct orders for the young Mage to be executed. (orders which were revealed to have been fabricated by corrupt humans.)
However, it seems that the humans of Edolas do not truly wish to be led by the Exceed, as their King's ultimate plan, Code ETD, shows. In sending the Lacrima containing Earth Land's Mages on a collision course with Extalia, Faust plans to unleash an unlimited amount of Magic Power to sustain his kingdom for all eternity and by doing so, he has declared war on the Exceed race.
Apparently the Queen's God complex has even gone to her people's heads, as when Wendy Marvell and Carla try to warn them of their impending doom, they arrogantly disbelieve her and start throwing rocks at the "mere human" and "fallen angel".
However, everything, including the God Complex, the Queen's Magic Power, and governing the human race's death, were all just fabricated lies made up by the elders of Exceeds to protect their race. The Exceeds were a weak race that was often picked on by the humans. To protect their race, the Elders of the Exceed spread false information that the Exceed had many powers. This, in return, gave the Exceed more confidence to themselves while causing humans, even the skeptics, to fear the Exceed's godly power. The Exceed eventually added the Control of Human lives to their list of powers, but in truth they never had such power or wish to kill off any human. Shagotte was selected as the Queen because she had the ability to predict which human would die with perfect accuracy, and the Elders used that as evidence of the Exceed's power.
After Team Natsu and company returned back to Earth Land, it was explained by the Elders and Shagotte that there was no mission to kill the Dragon Slayers. Carla has the same powers of prediction of the future as Shagotte due to being her daughter. The reason why the 100 Exceed eggs were sent to Earth Land was because Shagotte had a vision of Exaltia crashing so she gathered 100 children from the Exceed and borrowed the human's Anima to transport them over. As not to start a widespread panic throughout the nation, the Elders said they sent them to Earth Land to assassinate the Dragon Slayers. After explaining this, the race of Exceed excluding Carla, Happy, and Panther Lily flew off to find the remaining children they sent over
One-hundred Exceed eggs, including Happy and Carla, were taken from their parents and sent to Earth Land on a mission to obliterate the Dragon Slayers. When Faust began using Anima to steal Magic Power from Earth Land, Queen Shagotte gave the Exceed in the field new orders: To capture the Dragon Slayers and bring them to Edolas. The Dragon Slayers would then be used to create a special type of Lacrima filled with Dragon Magic. Carla decides to abandon her mission to kill Wendy and brought her to Edolas without knowing that her mission had been changed to simply lure Wendy to Edolas. She becomes very distressed by the fact that she unknowingly betrayed Wendy and fulfilled her mission. Happy is unable to recall the details of his mission but refuses to be a puppet of the Exceed, making it clear that he and Carla are members of Fairy Tail first and Exceed second. After this, they escape and are declared outlaws.
This mission was later revealed to be a lie, due to the Queen's power she could see the events of the Edolas arc, and as such, she saw Extalia destroyed. So in order to keep the species alive, she sent 100 eggs to Earth Land to protect them, and lied about the mission to kill the Dragon Slayers to the other Exceed.