The Azure in BlazBlue, is located inside the Boundary and described as being the true power of the Boundary (or perhaps beyond the power of the Boundary). The Azure Grimoires are seemingly the only grimoires that have the power to tap into it. Yūki Terumi was the one who created the prototypes of the Azure Grimoires along with his improved Azure Grimoire he possesses. Ragna is able to access the True Azure through the use of his imitation Azure Grimoire along with the Idea Engine. Noel, unlike the other Prime Field Devices, didn't need the Azure Grimoire to become Kusanagi because she already inherited the Azure.
Arakune is obsessed with finding the energy of the Azure. He hunts down those that exude the same energy or traces of the Grimoire. Coming into contact with the energy is what caused his form to degrade into the monster that he is. During his story, he manages to warn Litchi that she is being devoured by the energy as well, and that she should resist the temptation to embrace it any further.
Interestingly, there appears to be different "colors" of the Azure as mentioned several times by Arakune and Litchi. (Or rather this can be inferred.) When Arakune has a flashback to when Haku-men pulled Ragna out of him; he mentions in great shock the Azure inside him being blue currently, and not black. When he meets Hazama in his true ending, he is unable to see the color of his Azure, simply making the remark along the lines of "Your color is...not able to be seen by me? Your color is? Where is it?" leading to the assumption Hazama's color of Azure can't be seen. Further more In Litchi's CS storyline, she meets Hazama and feels as though she is sinking into a pit of darkness and despair, afterwards saying this was the Azure. Later when she tends to Ragna, she has another episode and feels a different presence of it; a warm blue light that seems inviting. Later Torakaka pulls her back into it by saying not to let herself be "seduced" by the Azure's power. Noel's Azure also is "warm and beautiful", due to having inherited the True Azure.