The White Council
The White Council is a group of Eldar Lords and Wizards of Middle-earth, which Galadriel initiated to contest the growing power of Dol Guldur. For the era of Lotro it is a successor to the original White Council of the Second Age, and is known as the Second White Council.
Superficially they resemble Men but as Maiar they possess much greater physical and mental power. Maiar are the same kind of spirits as the Valar, but weaker. Physically they were "real" Men with the feelings of urges, pleasures and fears of flesh and blood. This was for good and bad, Gandalf chose the righteous side while that cannot be said of Saruman. Few knew who the wizards really were, most thought they were Elves or wise Men.
The first three wizards were Saruman, Gandalf, and Radagast the Brown ("man of skill", "elf of the staff" or "wand-elf", and "tender of beasts" respectively); two other wizards are traditionally known as Alatar and Pallando. As the chief of the five Saruman used a white robe, while Gandalf used a grey, Radagast a brown, and Alatar and Pallando used sea-blue robes. Only Gandalf and Saruman play important roles in Lotro, Radagast appears briefly, but the blue wizards not at all.
The five wizards came to the Grey Havens in the north-west of the Middle-earth about 2000 years before Frodo set out on his quest to destroy the One Ring and Sauron was defeated. Already when they arrived they appeared as old because the Valar hoped the inhabitants of Middle-earth would accept their wisdom of age.
The Valar rarely meddled with the Middle-earth but they did not forsake the faithful Men and Elves, thus they sent the wizards. Rather than direct force of the events, and repeating earlier mistakes, they hoped instead to help Men achieve their own destiny. This is why the wizards were landed within the domains of Men, the Valar anticipated that Arnor and Gondor were in the best prerequisites to resist and overcome Sauron.
First White Council
The First White Council was formed after Sauron's defeat in Eriador in SA 1701, before the wizards were sent by the Valar. Little more is known about the council than it was led by Gil-galad and its decision to abandon Eregion in favour of Imladris.
Second White Council
The Second White Council added the wizards Saruman and Gandalf to the board. Also the chiefs and rulers were added, including Lady Galadriel of Lothlórien, Master Elrond of Rivendell, Círdan the Shipwright of the Grey Havens, and a few more. Against the wishes of Galadriel Saruman was appointed the head of the council.
One of the milestones of the Second White Council is how it handled the situation when Gandalf learned the identity of the Necromancer, who ruled over Dol Guldur. In a meeting on TA 2851 Saruman dissuaded the others to ignore Gandalf's discovery and refrain from action. However, realizing that the One Ring was possibly still around and that Sauron actively searched for it, Saruman relented and in TA 2941 Sauron was defeated. It was the meeting for these happenings that drew Gandalf away from accompanying Bilbo Baggins, Thorin and the others during their journey to the Lonely Mountain, as told in The Hobbit.
Another important meeting was in TA 2953 to discuss the One Ring. It was well known that Sauron was searching for it but Saruman insisted that it had fallen into the water and had been swept out to sea. Gandalf already suspected that Saruman desired the One Ring for himself and there was much tension between the two.
Of the five wizards who the Valar sent to the Middle-earth the outcome were quite different. Gandalf remained true to his quest to use his great wisdom to inspire to actions which would achieve Men's own goals. He nourished an affection for the Hobbits of the Shire without losing sight of his mission to let Men establish their reign. Eventually, and in times after Lotro, Gandalf went over the Sea to Aman along with the Ring-bearers and many of the Elves and so he left the Middle-earth.
Saruman nourished a desire to set himself up as commander in an attempt to contest and defeat Sauron. Originally he was the most powerful wizard. However, in TA 2953 Saruman managed to obtain control over Isengard and Orthanc; this made it possible for him to deepen his study of the lore of the Rings of Power and gradually he became influenced by Sauron via the palantír of Orthanc. Finally the Ents broke down Isengard, Gandalf broke his staff and expelled him from the White Council, and Saruman was killed by his servant Gríma Wormtongue at Bag End in the Shire.