EDIT: HEAVY SPOILERS BY THE WAY
Haven't read any of the other responses. Glancing at Roberto's, that's probably the best answer so far. Here's my extremely simplified version.
On Metal Gear, Snake is sent to destroy Outer Heaven. On a twist of events, his master, Big Boss, turns out to be the OH commandant. Snake wins.
On Metal Gear 2: Solid Snake, Snake is sent to Zanzibar Land. Here he faces the most fierces of enemies, Gray Fox, after discovering that again, Big Boss was behind everything. Using a cigar and a lighter, he sets BB on fire and destroys him forever.
On Metal Gear Solid, Snake is sent to Shadow Moses, where his brother - Liquid Snake - is planning to launch a nuclear strike and bring back Outer Heaven. Part of Snake's team betrays him, like Naomi Hunter, who was resented that Snake killed her brother - Gray Fox - and thus infected him with a virus called FOXDIE, which only aims to kill all of Big Boss' sons. That's right, Snake is son of Big Boss. Eventually, Snake meets Meryl, his comandant's daughter, and Otacon, a clever otaku scientist that designed Metal Gear, and the three end up Liquid's regime. Metal Gear, by the way, is a bipedal weapon capable of nuclear strikes and armed with a rail-gun.
On Metal Gear Solid 2,Snake, teaming up with Otacon (and now War Criminals after Shadow Moses), Snake infiltrates an USS tanker which apparently has a Metal Gear. Snake's mission now is to destroy every single metal gear in the world, after the blueprints were leaked on the Shadow Moses incident. Revolver Ocelot, a Liquid comrade that lost the arm during MGS1 events reinplanted in France Liquid's arm to his body, and now Liquid takes control of his mind - from time to time -. Ocelot voids Snake plans of destroying the Metal Gear and thus sinks the USS tanker escaping.
Years later, a soldier by the name of Raiden is sent to Big Shell, a facility in the middle of the ocean were the president of the United States is being held hostage. This mission is actually a farce enforced by the mistery group known as The Patriots; a formless, shapeless entity which aims to control US digitally, by creating context and censoring information. The whole Big Shell operation, which mimics the same exactly events of Shadow Moses, serves the Patriots to analize every single angle needed in order to evaluate all the possibilites and thus began control of America, and the world itself. At the end, a big Metal Gear crashes into Manhattan, and Raiden kills his former master, Solidus Snake, which is Big Boss third and last son. Snake congratulates Raiden at the end, and tells him to look for his own path in life now that he's free, after being a puppet to Solidus and THe Patriots all his life.
On Metal Gear Solid 3, we actually go to the past, to the days were Big Boss was a mere soldier. During the Cold War, Big Boss was sent to the URSS to kill his former master, The Boss, which is now a war criminal, and put an end to Volgin's regime; a merciless war comandant who wishes to take control over URSS. After defeating both and coming back to America, Big Boss receives his codename Big Boss. Volgin's fortune (a lot of money he had in order to create his terror regime) is stolen by a younger Revolver Ocelot, which gives it to the original The Patriot members in order for them to create THe Patriots in the first place.
Somewhere here, also the Les Enfants Terrible takes place, in which Big Boss had his cells cloned and three sons are created: Solid Snake, Liquid Snake and Solidus Snake.
On Metal Gear Solid 4, Solid Snake is now an old dude due to accelerating age of his cells. He has one last mission, to defeat Liquid again, after he has taken control of Revolver Ocelot's mind completely. Considering Liquid wants to destroy The Patriots, The Patriots help Snake and his team (Otacon, Meryl, Raiden and the others) in order to foil Liquid's plans. Snake eventually manages to put out not only Liquid, but also The Patriots who inadvertely let their guard down. After everyone happily lives after, Snake tries to commit suicide as FOXDIE, implanted by Naomi Hunter, has corrupted itself and now threatens to spread to every single individue in the world.
Snake is unable to kill himself, and out of nowhere, Big Boss appears. He tells Snake that he won't fight him, that he was mistaken and Outer Heaven should have never existed in the first place. Then he also tells Snake that his FOXDIE won't kill anyone, so he doesn't have to worry because the world is safe. Then, Big Boss dies from Snake's FOXDIE.
The end.