Shinra, Inc. is a huge, superpowerful megacorporation that essentially rules the world. The base of operations is the huge Shinra Building, located in the middle of the most heavily populated (and polluted) city in the world, Midgar. This company has their own private army and a super powerful elite divsion of troops called Soldier. They are the masterminds behind everything. Nothing in the world goes down without Shinra's knowledge. They figured out how to harness Mako Energy from the Life Stream by way of Mako Reactors across the world. Everything is powered by them. It was discovered, however, that Mako Reactors are sucking away the planet's life energy. And if the reactors deplete all of the available Mako Energy, the planet will perish. This is why the small rebel group known as Avalanche has taken to blowing up Mako Reactors.
President Shinra was the president of Shinra, Inc....until Sephiroth used his body as a pin cushion for his Masamune. He was pretty ruthless and would stop at nothing to get what he wanted. He wanted to have Dr. Hojo experiment on Aeris, so he kidnapped her. It was shortly after that, that he met his demise. You never fight him, but you do get to see his blood all over the place.
After his father's death at Sephiroth's hands, Rufus ascended to the presidency of the company, which basically meant that he ruled the world. Rufus doesn't exactly have the business sense of his father, so he figured he'd rule by fear. Like his father, he took what he wanted. He's also a double-crossing no-good swindler. He'll help you one moment, then when he no longer needs you, you find that you'll be the subject of a public execution. My, isn't he a nice guy? You do fight him, but very early in the game, just after his father's death. You fight him alone as Cloud. He's not hard, but he's accompanied by his pet doggie. Hit the dog FIRST, then concentrate on Rufus because the dog will put a shield up on Rufus, which'll make him harder to beat. You don't have a lot at your disposal at this point in the game, but hit him with Poison and Lighting while making sure to heal yourself. When he's had enough, he'll get on his helicopter and flee like the coward he is. And he'll indirectly give you a lot more trouble throughout the rest of the game. But don't worry. Though he thinks he's safe in his cozy little office on the top floor of the Shinra Building, something's gonna hit him when he least expects it. I know, you'd like the opportunity to off this weasel yourself, but oh well...