I apologize in advance for any rambly-ness. I started this at about 4am this morning~!
First of all, it's probably important to start with what exactly the Geass is, so let's start with the Canon facts that we know:
 It is bestowed to people in the form of a contract. (1)
 The Geass manifests in different forms for different people.
 The power of the Geass grows with use. Those unable to control it are eventually consumed by it. (2)
 Those that bestow the power of the Geass cannot be affected by it. (2)
 It is possible to fight the affects of the Geass, or break free of its power. (3)
 The power of the Geass is somehow connected to the exsistance of humankind, and it may be used to destroy just about anything. (4)
 There have been 4 known Geass users.
 The power of Geass manifests in a person's eye(s).
 Geass makes a person different than other people, seperating them from others. (5)
Now, this is what we know so far. If there's anything I missed, please let me know. ♥ I haven't listed the individual stipulations and idiosyncrasies for each Geass user, 'cause then we'd be here forever...
Aside from what we know, there's also a few things that are suggested in the anime:
 There is a connection between Geass and the following:
 Those that can grant the power of Geass also seem to have other powers, including the ability to affect a person's psyche. (10)
 Artifacts and archeological sites related to the Geass are distributed throughout the world, some of them containing traps known as "thought traps" (11)
 Although "geass" is the official term for the power, the basis for where this word comes from is not given to us.
Aside from these details, (and probably a few others I might've missed), we are only able to speculate on the nature of the Geass. Of course, it's been brought up several times that the term "geass" is a Japanese deriviation of geis or geas. Below is the information I dug up on what a "geis" is:
According to timelessmyths.com:
The word geis means "bond", a prohibition, taboo or injunction. The geis is tied with one's fate or destiny. The violation of one's geis, will lead to some misfortunes, and in most cases, to one's own death. They were something like a curse or a blessing. The Ulster Cycle stressed the importance of not breaking his or her geis, yet it seemed to be unavoidable.
Cu Chulainn had a geis where he was not allowed to eat meat of a hound (because he was named as a hound), but the hero was tricked into breaking his geis.
In the tale of the Destruction of Da Derga's Hostel, the high king, Conaire Mor, has an interesting number of geis impose upon him. Conaire was warned of not killing any bird, because his father (Danann) could shift-shape into a bird. One by one, Conaire broke each geis. When he broke one geis, he set the chain reaction where he will break all of them in rapid succession.
However, there are occasion when avoiding violation of the geis, does not necessarily means that diaster will be avoided. Avoid breaking a geis can sometimes work against the person. There are several famous tales, where the heroes met their death had avoided violation of their geis.
I[t is] not really certain, how a person find out what his or her own geis. Just about anyone can impose a geis upon another. Sometimes a druid imposed a geis to a person, other times, by king, hero, and in several of the stories, they were imposed by the person from the Otherworld (a god or goddess - one of Tuatha Dé Danann). In fact the geis was probably linked to the Otherworld.
In the Welsh myths, there was something like the geis in Math Son of Mathonwy (Mabinogion), where Aranrhod imposed several curses (blessings) on her own son, Lleu. But unlike the Irish geis, these could be overcome. Gwydyon, brother of Aranrhod, helped her son overcome these curses, by duping Aranrhod.
Pwyll had to accept a couple of conditions set by Arawn, which including taking his place as Lord of Annwn, by being disguised as Arawn, and fighting Arawn's enemy, Havgan.
According to Wikipedia:
In Irish mythology and folklore, a geis (/ˈɡɛʃ/, plural geasa) is an idiosyncratic taboo, whether of obligation or prohibition, similar to being under a vow or spell.
A geis can be compared with a curse or, paradoxically, a gift. If someone under a geis violates the associated taboo, the infractor will suffer dishonour or even death. On the other hand, the observing of one's geasa is believed to bring power and good fortune. Often it is women who place geasa upon men. In some cases the woman turns out to be goddess or other sovereignty figure.
The geis is often a key device in hero tales, such as that of Cúchulainn in Irish mythology. Traditionally, the doom of the hero comes about due to their violation of their geis, either by accident, or by having multiple geasa and then being placed in a position where they have no option but to violate one geis in order to maintain another. [...]
A beneficial geis might involve a prophecy that a person would die in a particular way; the particulars of their death in the vision might be so bizarre that the person could then avoid their fate for many years.
So, now we have an idea of where Geass comes from, possibly, but was does all that have to do with stuff like Jupiter and the "Sword of Akasha"?
Well, according to Wikipedia (again), Akasha is the Sanskrit form of the word aether. This substance is known as many things, including 'the fifth element' or simply an 'idea.'
Akasha (or Akash, Ākāśa, आकाश) is the Sanskrit word meaning "aether" in both its elemental and mythological senses. In Hinduism, Akasha means the basis and essence of all things. The source of everything that exists. One of the Panchamahabhuta, or "five great elements"; its main characteristic is Shabda (sound). In Hindi the meaning of Akash is sky.
The word aether (αἰθήρ) in Homeric Greek means "pure, fresh air" or "clear sky", imagined in Greek mythology to be the pure essence where the gods lived and which they breathed, analogous to the aer breathed by mortals (also personified as a deity, Aether, the son of Erebus and Nyx). It corresponds to the concept of akasha in Hindu philosophy and is linked to Brihaspati (or the planet Jupiter) and the center direction of the compass. It is related to αἴθω "to incinerate", also intransitive "to burn, to shine" (related is the name Aithiopes (Ethiopians)), meaning "people with a burnt (black) visage".
-Bold tags added by me.
So, it's clear what the connection between Jupiter and the "Sword of Akasha" is, but how does this connect to Geis? When we look up what "Brihaspati" (or Jupiter) is on Wikipedia, we're told that Brihaspati is a god who represents the balance of past karma, religion, philosophy, knowledge and issues relating to offspring. He is concerned with education, teaching and the dispensation of knowledge. It's also important to note that those born under the influence of Jupiter are often fat, but their prosperity and empire are known to increase.
This doesn't make much sense if we interpret it that Charles and VV are setting out to destroy the very god that would actually support him. Brihaspati seems to be the type of god that Charles would worship and look to for wisdom, since he embodies so much that is part of Charles' life (including issues relating to offspring) This, however, is my speculation. Perhaps the desire stems more from VV's side, as a desire to erase past sins and past karma. By destroying Brihaspati, it would seem that Chaos would reign, and this is actually what Lelouch wishes to accomplish.
When looking at what Brihaspati relates to, again, we also see that he rules over the sign of Saggitarius, which is actually Lelouch's sign.
Perhaps what Charles and VV are setting out to do is bring down Lelouch? Perhaps, somehow, they knew ahead of time what Lelouch's goals and desires were, and in reaction decided to form a contract? If we say that Brihaspati is related to Lelouch, then it would make sense that Charles and VV are attempting to "destroy" him and his related god.
When we delve further into this thread of Hindu mythology and astrology, we learn that Brihaspati has an arch-nemesis known as Shukracharya (or Venus). Shukracharya is described as white and middle aged. He holds the Sanjivani mantra, which is a hymn said to revive the dead. He also has a daughter named Devyani who was rejected by the son of Brihaspati, but then later married Yayati, who founds to Kuru dynasty.
This reminds me a lot of Charles (and possibly VV as well). Throwing in the Sanjivani mantra, which he might've used with Clovis (remember when he says that he had spoken with Clovis and that Clovis was at peace on a beach?) or Shukracharya's daughter Devyani (who seems a lot like Euphie, to me, especially when you consider that she and her husband start the Kuru
But that still doesn't explain the connection between the Sword of Akasha and Geis. Then we throw Marianne into the mix. Remember her? It's hard not to, since she's Lelouch's mother. It seems that the more we find out about her, the more we learn that she might've had a hand in her own death. A martyr, possibly? Perhaps her death insured, somehow, that Lelouch would eventually meet CC? This, to me, seems likely when we consider that CC seems to be in contact with the supposedly dead Marianne.
This brings me back to CC, who is a fascinating character. One of the interesting things about her is the symbol that she bears on her forehead and also on her breast. This symbol also shows up on the ruins at Kaminejima. To me, this symbol is very similar to the astrological sign for Neptune combined with the symbol for Uranus. This seems to be yet another reference to astrology.
Neptune, like Jupiter and all the other planets of the Solar System, is named after a god of the Roman pantheon. Historically, it is believed that Galileo was the first to discover Neptune, but because he thought it was a fixed star in conjunction with Jupiter, he is not credited with the discovery.
Astrologically, Neptune is related to idealism and compassion, illusion, confusion and deception; with religions, spirituality and mysticism, the mass media, creativity in art and music, drugs, extreme sensitivity, fantasy and imagination, psychic phenomena and altered mental states. These descriptions seem to coincide a lot with the symptoms and affects of each Geis. According to Wikipedia, modern astrologers consider Neptune the ruler of Pisces, but prior to the discovery of Neptune, Jupiter the one considered as the ruler of Pisces. Even some modern astrologers consider Jupiter a co-ruler of Pisces. Some astrologers do not believe that Neptune rules any particular sign, even though they may use the planet in interpretation.
Uranus, much like Neptune and Jupiter, is a gas planet and in astrology represents the principles of genius, individuality, new and unconventional ideas, discoveries, electricity, inventions, and the beginnings of the industrial revolution. Uranus also governs societies, clubs and any group dedicated to humanitarian or progressive ideals.
This might mean that while we have yet to see Neptune and Uranus in the anime, they might have a relationship with CC, Jupiter, the cult of Geass, and the Sword of Akasha. Personally, I think that there might be a big correlation between them all that have yet to be explained.
But, what does all this tl;dr boil down to? Basically, we have two people, each with a Geis. With Charles, we know his contract with VV is to destroy 'the gods' using the "Sword of Akasha." Meanwhile, Lelouch has a contract with CC, but we don't know what that one entails. CC has yet to name her greatest desire. But somehow, it possibly has something to do with Marianne and possibly the planets Jupiter, Neptune, and Uranus.
With all this information in hand, what do you guys think? Personally, my head is swimming, but can you sort through it and come up with a possible answer?
(1) Season 1, Episodes 1, 14; Season 2, Episodes 5, Code Geass: Lost Colors
(2)Season 1, Episode 15
(3)Season 1, Episode 22; Season 2, Episode 1 and 2
(4)Newtype USA, September 2006 edition.
(5) Season 1, Episode 1 "If you enter this contract, you will live as a human, but also as one completely different. Different rules, different time, different life... The power of the king will make you lonely indeed. If you are prepared for that, then..."
(6) Season 1, Episode 1
(7) Season 1, Episode 19, 23
(8) Season 2, Episode 6
(9) Season 1, Episode 1
(10) Season 1, Episode 11
(11) Season 1, Episodes 19, 24