The Pure Grimoire (無垢なる魔道書 Mukunaru Madōsho), also known as No Name (ノーネーム Nō Nēmu), is a grimoire that turned Mai Hazuki into a girl as its vessel has to be female. Mai is the Pure Grimoire herself, since she is a Remix Heart.
Simply speaking, No Name is a Blank Grimoire (白紙の魔道書 Hakushi no Madōsho), that gains power of any code (grimoire) that is put into it. For example, with earth code it will gain earth powers, wind will give it wind powers, water gives water, and fire gives it fire powers. It can gain all sorts of power, including the power of the Blue, which will turn the Pure Grimoire into another BlazBlue.
In BlazBlue: Remix Heart, No Name was also known as the Nameless Grimoire (名もなき魔道書 Namonaki Madōsho). Before it fused with Mai, the grimoire was sentient and was capable of communicating with its vessel when she was unconscious.