The Timelord academy was a place where Timelords could go to further their education, in similar way to us going to college or university. Children as young as 8 began their learning process at the Academy. We learned in 'sound of drums' that Timelords spend centuries at the academy; during this time each Timelord looks into the Untempered Schism, which is an opening into time; "a gap in the fabric of reality from which can be seen the whole of the Vortex"The head of the Academy is the Lord Chancellor.
The Academy consists of several chapters:-
The Arcalian Chapter is the second most important of the chapters. Arcalians, as they were known, wore green and brown robes. It is not known for sure but its is believed this chapter was part of the Arcalian Council for Temporal Research; an organisation which existed on their home planet, Gallifrey.
This chapter was based on the The Green party. They had very little political power compared to other chapters. Their robes were a subdued light blue.
This chapter was based on the Social Democrats. The Dromeian robes were a silver-grey colour. Like the Ceruleans, the Dromeians had very little political power.
The Patrexes are an order of artists and aesthetes. They wore heliotrope robes. Unfortunately they lacked any true imagination and so were minor players in Time Lord politics. Part of their duty was to look as far into the future as they could. They were also responsible for the creation of the N-Forms, which were used in the war against the great vampires.
This was a chapter of devious Timelords. Not surprising to hear the Master and the Doctor were in it. The colours of this chapter was scarlet and orange. Rassilon, co-finder of the Timelords, originally headed the chapter even before the Timelords themselves. This has helped more Timelords from this chapter assume office of the president.
This chapter was of little power and their colours were quite subdued. They bankrupted themselves trying to build their side of Panopticon. Panopticon was the main room of the capital of Gallifrey.
As part of the introduction ceremony, all Timelords say:-I swear to protect the ancient Law of Gallifrey with all my might and brain. I will to the end of my days with justice and with honour temper my actions and my thoughts."
(Source Doctor Who wiki)