Wow, what a great question! Usually, whether or not a dye is permanent depends on how strongly it binds chemically to the material it is dying.

Permanent dyes, for example, usually bind covalently while non-permanent dyes bind ionically. So to understand why some dyes are permanent and others are not, you need to know what a covalent bond is and what an ionic bond is. When atoms, such as carbon and oxygen, bind covalently, they actually share some of their electrons. That is, when a carbon atom and an oxygen atom are brought close to each other, some of the electrons orbiting the carbon atom start spending some of their time orbiting the oxygen atom and vise versa. The sharing of electrons between the carbon and oxygen leads to a very strong bond between the two atoms. We call this kind of bond a covalent bond. The covalent bond is so strong that we say a new compound is formed, carbon monoxide, in the example we considered here.

Not all atoms bond covalently to each other. Some atoms form ionic bonds (or other kinds of bonds) which typically are much weaker than covalent bonds. In an ionic bond, the two atoms which bond to each other are electrically charged, one with a negative and the other with a positive charge. This might happen if the one atom essentially steals an electron from the other atom. The one that steals an electron becomes negatively charged and the one that loses an electron becomes positively charged. These oppositely charged atoms will attract each other (recall that oppositely charged things attract!) and form what is called an ionic bond. In this case, the electrons all remain close to one atom or the other; they do not share any electrons. While this bond can be fairly strong, it is not as strong as a covalent bond. Furthermore, such an ionic bond can often be weakened substantially by putting the atoms in water, causing the bond to break and the atoms to separate from each other.

When designing dyes, chemists often purposely include groups of atoms in dye molecules that will form covalent bonds with the material they wish to dye. For example, if they want to dye cotton or hair, they will make dye molecules that include groups of atoms which form strong covalent bonds with cotton or hair. If they want to have a less permanent dye, they will include groups of atoms that bond ionically to the cotton or hair they wish to dye.

So that's it. Thanks for the question.