Fallout 3's BoS or Capital Wasteland BoS have branched off from the original BoS, or respectfully FNV's BoS.
The BoS in FNV are dying out and only care about technology, where as Fallout 3's have "lost their mind" and decided to collect tech and save the wasteland. The Outcasts, are loyal to the original BoS's ideals and hate the CW BoS.
Basically:
Fallout 3: Capital Wasteland BoS - Those commanded by Elder Lyons and are the "White Knights". There are also the Outcasts who branched off of Lyon's BoS and are loyal to the F:NV BoS (Original) ideals. What they do to help the people of the wasteland are, protect them from the Enclave and Super Mutants, Raiders, Talon Company, protect caravans and manage Purified Water with an alliance with Rivet City's security force, as of Broken Steel.
Fallout: New Vegas: Original BoS - Original BoS, formally commanded by Elder McNamara. They are almost dead, and do not care about wasteland or it's beings. They send some members outside to collect technology and scout.
Outside those games...
There are The Midwestern Brotherhood of Steel. They are seen in Fallout: Tactics. They were sent on zeppelins out east, but due to a lightning storm, they crashed near Chicago. Those who survived, lost contact with the Original BoS.
They established bases outside/inside Chicago. They recruited wastelanders in exchange for food/supplies and what have you. Only Paladins get to wear the http://fallout.wikia.com/wiki/Power_armor_%28Fallout_Tactics%29. Recruits are assigned "regular" clothing. Also the switched around their http://images1.wikia.nocookie.net/fallout/images/thumb/c/c5/FOT_MWBoS_Logo.png/200px-FOT_MWBoS_Logo.png. It is just the sword and gears in reverse.
Eventually they branched off from the original BoS like Lyons, but in a
less stupid way (<--- OPINION ALERT!!!).
You can read more on the game http://fallout.wikia.com/wiki/Fallout_Tactics:_Brotherhood_of_Steel at the vault wiki and sort of branch off with their links to armor weapons, factions and such.
As for the BoS in Fallout 1 or 2, I probably shouldn't explain because I might get things wrong/mixed up, since the farthest I got in both games was to a group of plants, which unfortunately kicked my ass. Not saying I don't know the events, just haven't played the games, Fallout 1 and 2, enough to judge them accurately.
Then this game was made: http://en.wikipedia.org/wiki/Fallout:_Brotherhood_of_Steel
BUT do not familiar your self with it because it is a load of BULL [censored], not cannon, and BULL [censored]. Basically do not read up on it, play it, what have you. Do not judge the BoS based off that game.
Edit:
Long Term Goals (Respectfully):
Fallout 1 and 2: Technology Hoarding/Collecting.
Fallout: Tactics: Control, based off of the http://www.youtube.com/watch?v=u2ZcpcO7C58, which I sincerely hope is cannon. Fallout: Tactics hasn't really been announced cannon from what i have heard but Barnaky is the ending I hope will be.
Fallout 3, Lyon's BoS: Protect the Capital Wasteland and clear all Super Mutants.
Outcasts: Collect Technology, screw the wasteland.
Fallout: New Vegas: Collect Technology.