The Hearts of the Seven Princesses = The result of the surviving children from the Keyblade War using the light in their hearts to forge seven hearts of pure light. Those seven hearts were used to banish the consuming darkness behind a door (Door to Darkness) and thus that banished darkness became the realm of darkness. (Which is why the princesses are the only ones who can open the DTD, because their hearts created it).
The seven hearts of pure light were then used to restore the realm of light to what it is today: Countless smaller worlds seperated by invisible barriers. As such the seven hearts of pure light act as a substitute for the true Kingdom Hearts (which was lost in the darkness during the Keyblade War). Without those seven hearts there is nothing preventing the darkness from consuming the realm of light just as it nearly did at the end of the Keyblade War.
The Seven Guardians of Light: The original X-Blade was the twin and protector of the true Kingdom Hearts. Kingdom Hearts and the X-Blade coexisted alongside each other. During the Keyblade War the two elements that make up the X-Blade (Light and Darkness), were shattered into 20 pieces: 7 of light and 13 of Darkness. The light shattered into 7 fragments because of the fact that the 7 hearts of pure light now act as the substitute for the true Kingdom Hearts. So just like the X-Blade protected the true Kingdom Hearts, so too does the 7 fragments of light protect its subsitute: The 7 hearts of pure light now held by the Princesses.
In order to protect the princesses of heart, the X-Blade's 7 fragments of light each chooses a Keyblade Wielder to act as one of the defenders of the princesses. In a sense you can think of it like this: If we have the seven princesses, then the 7 guardians of light are the knights of the princesses. And the proof of their knighthood is the 7 fragments of the X-Blade's light.
------------------------------------------------------------------------------------------------
As for the 13 fragments of darkness that came from the X-Blade, we know much less about them. Now my theory is that Master Xehanort at some point in his life was chosen by one of the fragments of darkness from the X-Blade. As such MX believed that if he was worthy to hold a fragment of darkness, then by allowing a piece of his heart to mold and reshape the heart of another person then that person would also become likely to be selected by a fragment of the X-Blade's darkness. To this end he went about his plan to create the true Organization XIII: A group of 13 people who are either a form of himself (Xemnas, Ansem, Young Xehanort) or hold a piece of his heart inside them (Xigbar, Saix, Terra, and possibly Vanitas).
---------------------------------------------------------------------------
Now one of my theories about both the fragments of light and the fragments of darkness is that both sets of fragments share the same goal: To protect the realm of light.
But the two sets of fragments go about it in different ways. The fragments of light choose guardians to protect the princesses who uphold the realm of light currently. The fragments of darkness choose 13 people who seek to forge the complete X-Blade in order to bring the true Kingdom Hearts back to the realm of light where it rightly belongs.
-------------------------------------------------------
As for what the fragments of the X-Blade actually are. Well they are the light and darkness that make up the X-Blade, so I guess you would call them fragments of the X-Blade's "Inner key". Nomura once said that all Keyblades have an "outer key" which is the physical blade, and an "inner key" which is the part of the Keyblade that makes it so special, what gives it it's nature and powers. The X-Blade's light and darknesses probably are what make up the X-Blade's "inner key", since the X-Blade's nature is "The perfect crossing of light and darkness."
I think maybe the orb that MX used to summon the KH in BBS may have been his fragment of the X-Blade's darkness (which would explain how it summoned KH before Ventus and Vanitas forged their incomplete X-Blade) but this means the KH in BBS is the true KH but it is only a fraction of the true KH responding to only a fraction of the X-Blade.