Short answers, pulled from the story.
Kyubey's true purpose is to counter entropy and prevent the heat death of the universe. They belong to a hive-minded alien race called the Incubators who convert human emotions into energy, specifically targeting pubescent and prepubescent girls because they produce the highest energy yield.
Kyubey was created and designed by writer Gen Urobuchi. Producer Atsuhiro Iwakami described Kyubey as an epitome of the series' central theme of mixing cuteness with darkness.
When a girl makes a contract with Kyubey, her soul is extracted and placed inside a soul gem, while her body is rebuilt into a more resilient shell for fighting witches. Kyubey deliberately withholds this information until after the consequences become apparent.
Kyubey won the 1st Newtype Anime Awards for Best Mascot Character in 2011 and the Bronze Prize at the 2011 Net Buzzword awards for a popular catchphrase. The Nikkan Sports newspaper also gave Kyubey the 1st Nikkan Anime Grand Prix's Worst Dark Character award.
At the end of The Rebellion Story, Kyubey's plan to restore the old witch system fails when Homura takes Madoka's powers and enslaves the Incubators to absorb the curses of the new world. In the post-credits scene, Kyubey is shown on the ground, disheveled, shivering, and mentally scarred.
Anime News Network's Gabriella Ekens called Kyubey "the alien embodiment of utilitarian logic." Paste magazine ranked Kyubey the eighth-greatest anime villain, and Comic Book Resources placed the character first on their list of the twenty strongest alien species in anime.