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The Beginnings of Sephiroth
The creators of FFVII wanted to add a bit of mystery to the storyline. Unfortunately, they added too much, and many players were left with some strange ideas about many things, especially the villain, Sephiroth. I shall try to clear this up.

Many years before the events in the game take place, possibly 30 or more, Professor Gast was the leading scientist at Shinra, Inc. After uncovering a specimen which he believed to be an Ancient, or Cetra, he set about attempting to bring back the long extinct race by genetically altering humans by injecting into them the cells of this supposed Ancient, whom he called "Jenova". His research took him and two of his colleagues, Professor Hojo and a female scientist named Lucrecia to Nibelheim. During that time, Lucrecia became pregnant by Hojo. Hojo, seeing this as an opportunity to further Gast's research (and likely, his own career) persuaded Lucrecia to allow experiments to be done on the fetus growing inside her. These experiments, while not fully explained, most definitely consisted of the injection of Jenova cells. The amount was likely enormous, as this began seriously affecting not only the baby but the mother's health as well. The child borne to them was none other than Sephiroth. Lucrecia was never able to act as a mother to him. In fact, it's speculated that she died in childbirth, and the image Vincent speaks to in the game is her ghost.
Whatever may have happened to the mother, Sephiroth grew up, most likely being raised by Shin-Ra employees. He was trained to be the perfect soldier, and the first...well, SOLDIER. After proving his power, Shin-Ra began engineering more humans like him, trying to copy his power. They were only able to do this to an extent, likely because the injections occurred when they were already young adults and as such, they didn't absorb them as well. Sephiroth quickly became a celebrity in the eyes of most, and likely a monster in the eyes of some. He earned true fame after playing a key role in Shin-Ra's victory during the Wutaiin War.
After the war, Sephiroth took part in peacekeeping missions. One of these took place in Nibelheim. During this visit, Sephiroth uncovered paperwork regarding the experiments done on him before he was born. It's easy to understand why he went mad. The papers he uncovered, however, led him to believe that Jenova was not only a Cetra, but his mother. He attempted to retrieve her body from the reactor in Mt. Nibel, but was cut down by a very ticked-off low-level grunt by the name of Cloud Strife. Having been weakened, Sephiroth decided to simply decapitate his mum and take the head with him. Cloud attacked him yet again before he could escape, and was impaled by the Masamune. Now, here's where many might disagree with me. Due to a failure to properly evaluate the threat posed to him, Sephiroth, along with Jenova's head, was thrown into the mako pool below, and his body was carried to the Northern Crater, via the Lifestream. Note that I said 'body'. Sephiroth died 5 years before the game. The figures seen throughout the game are not Sephiroth, or Jenova, or Sephiroth-clones. Rather, they are projections created by Sephiroth.
You see, while Sephiroth was very much dead, his will was simply too strong to allow itself to be lost in the Lifestream like everyone else. It went on being Sephiroth, even though his body was dead and encased in a crystalline cocoon in the bottom of the Northern Crater. He wanted power. That much is evident. He also needed a body. Manipulating others just wasn't cutting it, as they sometimes fought back, and he likely planned to use the power gathered by sucking the planet dry of spirit energy to create some sort of physical body. This, along with destroying the planet in order to take revenge for his "mother", Jenova, was his purpose in summoning Meteor. Things did not go as planned, however. He was defeated, and weakened to a point that, without a physical body, there would be no way he could properly control the Lifestream as he wished. That is why, at the very end, Sephiroth confronts Cloud in his mind: he is, literally, fighting for control over Cloud's body.
In case I haven't cleared up every rumor with my explanation, here are some other things I've heard:
--Sephiroth is human. An insanely strong human, but human still. He had 2 very mortal, very human parents. Though he says that he intends to become, "a god to rule over every human soul," I believe this was more of a metaphor than anything. Regardless of what he meant by it, he obviously never accomplishes this, despite the fact that the name of the music playing during the Bizarro Sephiroth fight is called, "Birth of a God". This might have fueled the belief that Sephiroth has already achieved divinity.
--Jenova is NOT Sephiroth's mother, any more than Vincent is Sephiroth's father. While these two characters do play an important role in his life, they are not actually related to him by blood.
If I missed anything else, please tell me. I am aware that you may not agree with some of the things I mentioned here. I assure you that there is evidence from the game that supports my claims. But, these are simply the best theories I could come up with.

I agree, very well written. It was a nice summary.

Um, thanks...y'all. I would've replied earlier, except Gaia was being odd with the replying thing...
I'd been meaning to write something like this and post it somewhere. And I was given the go-ahead on here, so I figured, why not? Thanks for the comments...if you see anything I can improve, please tell me 3nodding