Grimoire Heart was one of the top three Dark Guilds that the Balam Alliance is comprised of, as well as the one commonly considered the strongest Dark Guild. The guild owes its name to the Devil's Heart, an organic device which is the source of Guild Master, Hades' immense power, kept inside the guild's airship.
When Grimiore Heart first appeared in the story, their goal was to acquire all the keys rumored to be used to unlock Zeref's seal, as well as to locate the Black Wizard himself. Once these goals had been achieved, their intention was to use him, discovered in a slumbering state on Tenrou Island, to rule a world (referred to as the Ultimate Magic World) where the 90% of the world's Humans, those that cannot use Magic, would not be able to survive. Due to the Fairy Tail guild's presence on Tenrou Island when Grimoire Heart decided to make the move to unseal Zeref, one of its goals was also to get rid of "Fiore's #1 guild" in order to reach Zeref; a goal which, before finding all of the keys, they were also willing to leave to others, such as the fellow member of the Balam Alliance Oración Seis.
Grimoire Heart doesn't seem to have any stable location. Its members live and travel on a giant airship that flies high above the land. Because of this, it is an elusive guild. Among other things, the airship housed the Devil's Heart, Master Hades' source of power.
What is known of Grimoire Heart is that its founder and original Guild Master, Hades (real name Precht), was the Second Guild Master of Fairy Tail, and was made such by Fairy Tail's founder, Mavis Vermilion. In year X736, he bestowed the title of Third on Makarov Dreyar so that he could go on a journey. On this journey, Hades traced back the origins of Magic to Zeref, and saw what he referred to as the "essence of Magic". It is assumed that he turned "evil" after this point, and that he created Grimoire Heart and became its Guild Master in order to awaken Zeref and make the Ultimate Magic World real. Hades proceeded to gather young boys and girls with strong Magic potential, such as Ultear Milkovich, and spent years raising and training them for them to become the Seven Kin of Purgatory, some of the greatest Mages under his command. These Mages use a variety of different types of Magic that are as close as possible to the source of all Magic, taught to them by Hades himself: Lost Magic.
As a member of the Balam Alliance, Grimoire Heart is acknowledged as one of the three most powerful Dark Guilds in Earth Land. Even among them, it was stated to be the strongest one by Levy McGarden, and one of the three greatest protuberances of the Magic world (alongside Fairy Tail and Zeref) by Lahar. Unlike the Oración Seis, Grimoire Heart has far more than six members, having several core members as well as numerous members of lesser rank. Many of its most prominent Mages have been shown to be incredibly powerful, and when they arrived to fight Fairy Tail on Tenrou Island they managed to battle and even defeat some of its top Mages. Its Guild Master, Hades, was able to defeat Makarov Dreyar, Fairy Tail's Master and one of the Ten Wizard Saints, with ease, and to similarly keep the upper hand over both Fairy Tail's Team Natsu and excommunicated S-Class Mage Laxus Dreyar, eventually being defeated by them just due to external intervention. The Deputy Commander of the guild, Bluenote Stinger, was able to fight on almost equal grounds with Gildarts Clive, Fairy Tail's Ace, and all members of its strongest team (the Seven Kin of Purgatory) have proven themselves to be of S-Class level strength. Azuma, in particular, even managed to combat powerful Fairy Tail's S-Class Mages the likes of Mirajane Strauss and Erza Scarlet on equal grounds. Even two "minor" members such as Yomazu and Kawazu, through their combined efforts, were shown keeping up with an S-Class level Mage like Gajeel Redfox for a certain amount of time.
After their defeat at Fairy Tail's hands, Grimoire Heart's strength lowered significantly: three of their top members, including Hades himself, were killed, one was turned into trees, and two (technically three, seeing as Caprico was being possessed by Zoldeo) of them defected from the guild. Following Tenrou Island's destruction at the hands of Acnologia, Grimoire Heart had disbanded all together.
Following the defeat of Grimoire Heart and Acnologia's destruction of Tenrou Island, the surviving members had gone missing. Even after seven years have last, the current whereabouts and the statuses of the surviving members of Grimoire Heart remain unknown. The only exception is that of Rustyrose who seeks to revive the guild (though that's anime exclusive).