Note: Giegue is Giygas. Nintendo is just inconsistent in their translations (for example, in Brawl Pokey is now "Porky" ((his original name). The true translation of his name is Giyig, but this was never used... so the link between the characters is often debated... though in Japan, it's pretty clear. Giyig was the final boss in Mother.. and Mother 2 had the title "Giyig strikes back". I would also like to point out that Giygas's original design in Mother is what inspired Mew and Mewtwo... although "inspired" might be stretching a bit. The developers of the first two games of the series was Ape Inc.... which was later renamed Creatures Inc... of Pokemon fame. One of the two lead composers of the Mother series is the current president of the company.
The original Giygas in Mother (final fight)Mother: Early 1900s: Aliens abduct a young couple, George and Maria, to study humans in hopes of making peaceful contact. Giygas was born around this time, and in good faith, they decide to let this couple raise him. They loved him, and memories of his childhood proved to be his only weakness, though singing in general seemed to have always cause him pain, as Maria recalled.
"I loved him as if he was my own child. He was always wagging his tail just like a pup... EXCEPT for when I tried singing lullabies to him"
Mother: Early 1900s: Sometime later: George has been secretly studying the aliens in return... and he somehow picked up on their psychic abilities. After this was revealed... the alien race felt betrayed, and decided that humans now posed a threat. At some point Maria trapped herself inside her own magicant (a dream like world created by one's own mind... one that can be used as a means to imprison/ banish someone), while George stayed on earth raising what was left of his family, and spreading his research of PK (Psycho Kinetic) abilities. Giygas felt especially betrayed by this, and this fueled his campaign to rise to the top of the ranks of his society... eventually becoming his race's leader.
Mother: 1980s The invasion begins. Ninten, the great grandson of George and Maria, sets off on a journey to save the world. They eventually encounter Giygas and bring him down by singing the lullaby Maria wrote for him.
The game ends here, but not before a cliffhanger of a man on the phone telling you that the war wasn't over yet.
Mother (As explained by buzz buzz in Mother 2): 1990s: Giygas returns to in a plot that parallels what happened in Mother 2, only he succeeded in his invasion this time and his race set off to conquer the rest of the universe.
Mother: 2000s (semi-theory, as a few bits are unclear in the games): At some point Buzz Buzz's people defeated Giygas by trapping him in his own magicant and banishing him to the beginning of time... leaving him in his most vulnerable state.
This is where the time line changes. Giygas no longer invades in 1980 and again in 1990 with the rest of his race.
Mother 2: Beginning of time: Giygas eventually frees himself from his own magicant by using his immense power to shed his mind and body
Mother 2: 1990s: The damage done by Giygas in the original timeline in the 2000s was too severe to salvage, so Buzz Buzz's people send him ten years into the past in the body of a bee to warn Ness about the events to come... though his story was cut short by Pokey's mother swatting at him thinking he was a fly (not really a spoiler as this happens at the begining of Mother 2). Giygas attempts to invade the earth again, only this time in a different manner. In Mother he did it by using science and technology using his mothership as a base of operations. In Mother 2, there is no alien race. Giygas is alone in his work... and he does it by "infecting" the world with his presence... since he can no longer directly influence the world in his current state (being nothing more than his former self's subconscious). Though... the technology used in the first (like the Starmen for example) remains. There is no way to defeat him because he is everywhere and nowhere. The invasion was a long time in the making, and the only way to stop it is to prevent it from happening. The only option is to travel back to the beginning of time... which is what the heroes of Mother 2 set off to do.