The oath of the Blue Lantern Corps is filled with hope, a message they spread across the universe while defending it from threats of all kind. The Blue Lantern Corps was formed by Ganthet and Sayd, two former Guardians of the Universe who were rejected by their brethren after embracing emotion and falling in love with each other.
In the Book of Oa, when mentioning the prophecy of the Blackest Night, a passage reads "Elsewhere, a flicker of hope will shine from deep space, like a lighthouse warning the ships away from the rocks. The blue light will hold the line in spirit if not in strength." Realizing that this blue light would play a crucial part in stopping the oncoming crisis, the two of them worked together to create a blue ring and create a Corps fueled by hope, adopting the motto "All will be well".
Arguably, a Blue Lantern is the least powerful out of all the Corps, at least on their own. Its minimal capabilities include flight and a protective aura, as all Power Rings have. A unique ability it has is that it can scan a target and create a projection with the effect of calming the mind or bringing hope. It also has reasonably powerful healing abilities, with a projection once being able to regrow Sinestro Corps member Arkillo's entire tongue back. The ring can only really be used offensively, however, within the presence of a Green Lantern, as hope is useless without the willpower to act upon it. When this is the case, the ring is capable of creating a much more deadly energy blast, and the close proximity of the two will boost the power levels of all Lantern rings in the immediate vicinity to above their usual threshold.
In terms of membership, the Blue Lanterns are one of the most exclusive Corps, with maybe the exception of the Orange Lanterns solitary member, Larfleeze. While Ganthet and Sayd are the creators of the Corps, the true leader of the Blue Lanterns is Saint Walker, an alien priest from a planet with a dying sun who gave his people hope for survival. This hope attracted a blue ring, which he used to de-age the sun, saving his race from extinction. He joined the two former Guardians, and recruited a few others to their cause, such as Brother Warth, an elephant-like alien, and Brother Eesok, a humanoid creature with flaming hair. There usually seems to be no more than ten Blue Lanterns at a time, as strong hope is scarce in the universe.
A few other heroes have taken up the blue light temporarily, with Green Lanterns Hal Jordan, Guy Gardner, and Kyle Rayner all harnessing the power of hope for a short while. Perhaps most notably was during the "Blackest Night", when Ganthet was able to make a duplicate for each ring color, a blue ring was gifted to Barry Allen, who harnessed both the speed-force and the emotional spectrum.
Sadly, there is little left of the Blue Lanterns now. A being called Relic swore to destroy all "lightsmiths" in a twisted way to protect the universe, he attacked the Blue Lantern Corps, killing every member except for Saint Walker. He lost his ring temporarily after losing all hope, until Kyle Rayner managed to reignite it through his actions as the White Lantern. Since then, he has appeared sporadically, and without the whole Corps behind him, and it is unclear when we will next see the army fueled by the blue power of hope.