'Verte', or green absinthe gets it's colour by infusing plants in the scented alcohol distillate after the distillation. The plants used in this step vary and each will add flavour as well as colour. Colouration is a very skilled process (this is mostly why the majority of absinthes today are artificially coloured), if the distillate stays in contact with the botanicals for too long the flavour balance along with the colouration of the absinthe can be spoilt. The natural colour can also change and develop as the absinthe ages, most taking on a yellow-green or 'dead-leaf' colour. The colouring step may also create a slight deposit which can be found in the bottom of the bottle - this is not a defect, moreso a sign of quality, natural manufacture.
Absinthe 'blanche', white or colourless absinthe, known as 'la Bleue' in Switzerland, is distilled and then left without a colouring step. It will not necessarily be the same as an uncoloured absinthe verte though - to add complexity without colour, there are usually more plants used when distilling a blanche.