Here's the origin story behind the Faceless Men in Game of Thrones, including what they were really after in the series. The Faceless Men is a guild of religious assassins based in the Free City of Braavos. They call the House of Black and White their home as they work as servants of the Many-Faced God, a.k.a. the God of Death. The memorable member of the Faceless Men in the HBO series was Jaqen H'ghar (Tom Wlaschiha). For a short time, Jaqen trained Arya Stark (Maisie Williams) to become "no one", while teaching her the beliefs of this specific society.
Much of the Faceless Men's history and their overall narrative was left out of the Game of Thrones TV series. Many viewers expected Jaqen and the guild of assassins to be part of the bigger picture but that wasn't the case. The Faceless Men were a much more prominent segment of characters in George R.R. Martin's A Song of Ice and Fire. In the novels, the Faceless Men were better portrayed as a religious cult. More members of the guild were introduced, and not just in Braavos. The last book published, A Dance With Dragons, saw Arya gearing up to start her first apprenticeship with the Faceless Men. It seems as though the books won't push aside the Faceless Men the same way the show did.
The Faceless Men were known to give up their identities to become "no one" while living by the motto, "Valar Morghulis" or "All Men Must Die." They were trained to deliver the "gift" of death even if that meant wearing the face of another person in their mission. The origins of the Faceless Men predated the Doom of Valyria. The society was founded by the slaves that worked in mines under the Fourteen Fires, a chain of volcanoes in Valyria. The men prayed to their different gods but they soon realized the fundamentals of their religions were the same. Over time, the slaves began worshipping the god "with a hundred different faces" who was later known as the Many-Faced God. The killing started when the Faceless Men brought the gift of death to fellow men who were suffering in the mines. The unknown first Faceless Men once brought a "gift" to the Valyrian masters and some believe it resulted in the Doom of Valyria, suggesting that too many mages casting the spells to control the Fourteen Fires were killed.
The surviving Faceless Men relocated to Braavos after the cataclysmic event. They then reportedly charged exorbitant fees for their services. Their driving force, however, was a bit ambiguous. It was never truly made clear what they wanted they wanted to accomplish. Some believed that the Faceless Men wanted to bring another Doom of Valyria to Dragonstone with the use of a dragon egg. This followed with the notion that they were responsible for the deadly event in Valyria centuries before.
Another theory suggested that the Faceless Men were allies of the Others, also known as the White Walkers. Being a death cult, it was thought that the Faceless Men might have had sights to taking over and exterminating humanity. The construction of the Wall was detrimental to those efforts and the Faceless Men were thought to have plans to help destroy the vast structure. They needed dragonfire which could explain why many of the dragons in Westeros disappeared.
A third explanation into the goal of the Faceless Men in Game of Thrones was their desire to simply rid the world of magic and dragons. As slaves in the Valyrian mines, the Faceless Men were targets of abuse by dragonlords and the mystical beasts. There was thought that the Faceless Men wanted to prevent that magic from being a prominent force in the world. This could link to why Jaqen snuck into the Citadel in the books, which remains a plot point to be developed in The Winds of Winter. He may be looking up key information centered on dragons to assist in their efforts. Game of Thrones lost their chance at giving the Faceless Men a worthwhile story, but the book series still has an opportunity to give insight into the guild's endgame.