First off, both groups started out orginally in the west. The Enclave being the remnants of the American Government (their original HQ was control station ENCLAVE, an Oil Rig hundreds of miles off the west coast and where they had retreated to before the Great War), and the Brotherhood started at Mariposa when U.S. soldiers declared their succession a few days before the Great War began. The Enclave's Oil Rig was destroyed during the events of Fallout 2, and they fled east under the orders of the new President, John Henry Eden, establishing themselves at Raven Rock.
During Fallout 2, their interaction was fairly minimal, with the Brotherhood mainly wishing to try to steal Enclave tech and keep a low profile. Once the Oil Rig fell, the Brotherhood eventually helped the NCR to overrun an Enclave base at Navarro. The Enclave on the other hand, has Sebor stated, had plans to wipe the earth nearly clean of life with the use of the "FEV Curling 13" virus which they were going to pump into the Earth's jetstreams at the Oil Rig.
The Enclave and BOS are fighting in Fallout 3 primarily because of the purifier. Two Enclave leaders have plans that revolve around using it somehow (Autumn wants to control distribution of water, Eden wants to use it to distribute a deadly toxen to purify the wasteland), and the Brotherhood won't allow the Enclave to keep hold of the purifer, and wishes to use it to distribute water to the wastelanders.
I'll go ahead and throw in the reason's I personally support the Enclave:
These are the main reasons I support the Enclave:
1. Whats in a name? Personally their title "The Enclave" brilliantly sums up something I love about them. The fact that they are indeed an enclave, in the definition of the word, which means that they are a isolated group in the midst of a "whole lotta other crap" (that’s the technical term for the wasteland btw). The fact that they are isolationist is one thing that drew me to them. Their strategy, in my view, is amazing. They sit back and work on building their strength while waiting for the opportune moment to strike when everyone else is fighting and weakening each other. Brilliant.
2. Technology and Military Power: They are without a doubt the most advanced faction in the game universe. Their ability to develop and create new weapons, defenses, robots, and even the only aircraft in the world (the Verti-birds) makes me feel like I'm supporting a faction that knows what its doing in terms of actually being able to rebuild the ruined wastes. While others sit in squalor and twiddle their thumbs like morons (looking at you BoS), the Enclave has tech that they could never even come close to matching.
Their military strength is also something that is a sight to behold. As I said before, they are the only faction with the ability to have air-superiority and advanced logistics and mobility. Let’s face it, had Lyon's or the Chosen one not been aided by bad writing, there is no real way that either of them would have succeeded in defeating the Enclave. (orbital strike targeting the citadel). Their advanced technology also factors into their strength as a military. They are the sole (I would guess besides any pre-war ones) source of plasma rifles and plasma based weaponry in the wasteland. Maybe not necessarily in the west (plasma in Fallout 1), but the East for certain. They are able to arm even the lowest ranked of their soldiers with a least a laser rifle and Advanced Power Armor and their strongest with Hellfire armor and an incinerator. Not even the high and mighty Brotherhood that everyone seems to like so much can match this, not even by a long shot. I also don't see their "lack of numbers" as a weakness. Indeed in some ways I see it as a strength.
3. Stars and Stipe’s Forever: They are without a doubt the United States of America. The Enclave of the United States of America is not just a fancy title you know. I like how the Enclave are also related to COG (Continuity of Government) plans and how they retain control over pre-war bases and bunkers (Raven Rock and Adams AFB for example).
4. Destroy, Rebuild, and Renew: Contrary to what a lot of people seem to think. The Enclave aren't in the business of genocide for the fun of it. They are restoring the old world by removing what they see as a grave threat to the future of the human race, mutants.