Their "quick history" seems a little inadequate. They should've at least mentioned Grey Fox.
Here, I'll write up a bit more, for anyone who's interested.
Spoilers for Metal Gear and MG2: Solid Snake, of course.
In the original Metal Gear, you play as "Solid Snake", a rookie member of Foxhound. Your personal hero, and the main agent of Foxhound, a soldier code-named "Grey Fox" was captured while investigating Outer Heaven, a military nation developing a nuclear arsenal called "Metal Gear". "Big Boss", a formerly legendary soldier and the commander/founder of your unit, orders you to infiltrate Outer Heaven to rescue Grey Fox, and ensure the success of his mission.
Along the way, your radio-contacted support team is mysteriously eliminated, one by one.
You eventually succeed in rescuing Grey Fox, some scientists, and making your way to the heart of Outer Heaven, where you find Big Boss. He had betrayed you, Grey Fox and all of Foxhound, in favor of his world-domination plans in Outer Heaven. He had sent a rookie like you into Outer Heaven, expecting you to die, not succeed in your mission.
You defeat Big Boss, and put an end to his plans. You and Grey Fox go home.
I believe that "Snake's Revenge: Metal Gear 2" on the NES was an action game, basically dealing with Snake taking out the guys who eliminated your support team, but I can't remember, because it's been a long time since I played it. Anyways, it's not considered the "real" sequel to Metal Gear, as it wasn't made by Hideo Kojima or anyone else from the first game.
In the real sequel, "Metal Gear 2: Solid Snake", which was only available in Japan, Big Boss apparently survived and retreated to Zanzibar, to rebuild a new army. For the final battle, the game apparently put you against Grey Fox, who had switched sides.
Metal Gear Solid, or The Twin Snakes as it's now being called, is the third game in the series.