Oh boy. Okay, let me try and keep it brief. Long story short, the first flame is the fire in the game that serves as a source of power in the series. It provides light in the universe, allows for Unkindled warriors to gain strength, and is the cause of a prolonged power struggle across all of the Dark Souls games. Warriors often go to the fire to link it, sacrificing themselves to keep it burning. The first flame is a bit of a double-edged sword; by the time of Dark Souls 3, it has been linked so many times that distortions in time and space have started to appear, and the world has fallen into decay. However, if the fire is put out, the world will fall into endless darkness. Not a very happy ending either way, although people generally seem to think that plunging the world into darkness is better than keeping the fire burning. Not sure why.