Dye ink is water-based. They are comparatively fast drying and meant to be used on surfaces that are porous like paper. These inks are water soluble and will run if mixed with water. It will inks wash off with water and a little soap, pretty easy to keep your stamps clean when using this type of ink.
Solvent inks are permanent inks that can be used to stamp on non porous surfaces such as glass and metal.
Pigment inks are usually richer in color and they take longer to dry. They are fade-resistant, and are good to use for embossing techniques
My choice of inks to use with copics is Memento.
If you have any questions please feel free to email me at firstname.lastname@example.org.