The Brotherhood of Legendaries (Or the Legendary Brotherhood) is a collective organization of legendaries collectively ranging from Arceus and Dialga to Jirachi and Mew. They are responsible for certain things such as the creation of new Pokemon, new species of life, new islands, and more. They also intervene in situations when things often get nasty in the mortal world. The Brotherhood is also responsible for eliminating threats created by members as well. For example, a mewtwo was found creating earthquakes in southern Zurandia. Two members were sent to surpress the threat and seal the mewtwo off permanently.
No one knows why the Brotherhood exists. However, one can speculate that they are in service of Arceus to handle daily errands as he, the omnipetant creator, often sits in a state of slumber to recharge his powers. Not all legendaries are part of the brotherhood and not all brotherhood members are strictly legendaries either. Over time, they have grown less and less active and the legendary Pokemon that used to thrive on Fierra have dwindled for some odd reason. Irregardless, they are still active, whether it would be some other dimension or in an isolated part of the world.