Big Boss was the first 'Snake,' codename Naked Snake. He was recognized as the greatest soldier of the 20th century, and while in a coma, some of his genetic information was taken to be used in cloning. He became Big Boss after defeating his mentor(The Boss) in combat, but felt betrayed by his country. He created Outer Heaven as a sanctuary for soldiers, where they would constantly be valued. At this point, he is the villain of Metal Gear 1.
Solid Snake is one of the products of the cloning of Big Boss. He has a similar appearance, but he has a different eye colour, and a totally different personality. In his rookie days, he worked under Big Boss, and was sent to Outer Heaven to eliminate Metal Gear. However, he later finds out that Big Boss is actually his enemy, and defeats him. He ends up going to Outer Heaven again, when Big Boss once again threatens the world with Metal Gear. He ends up killing Big Boss with a make shift flamethrower, put together with a lighter and aerosol can.
In MGS1, we're introduced to one of his brothers, Liquid Snake. Liquid is pretty much nothing like Solid Snake or Big Boss, outside of the fact that they share similar genes. He's killed by Snake, but lives on through his arm, which he used to possess Ocelot.
Solidus Snake is yet ANOTHER Big Boss clone. Former president of the US, and dedicated to the destruction of The Patriots, he serves as Raiden's father figure, and is ultimatly killed by him. Solidus' accelerated aging is taking place much faster then the other two Snakes.
The Snake in Brawl is not really Solid Snake, yet not Naked Snake either(and certainly not any of the other two), He's a combination of both Solid and Naked Snakes, with the outfit of MGS2 Solid Snake, yet a beard closer to Naked Snake, and also Naked Snake's blue eyes(Solid Snake has green). Brawl Snake is more of an almalgum of the two big Snakes, representing both past and present.