There are multiple scientific ways like any combination of Pigments, Chromatophores , Structural coloration and Bioluminescence by which different animals change their colour for several reasons according to specific environment and conditions.
https://youtu.be/Ihue9R8m6FA

Lots of Lizards ( Chameleon )have transparent out skin and multiple layers of skin inside which contains cells contained Chromatophores with natural Pigments melanin . Chameleon neurotransmitter signal when it's experiences changes in body temperature mood or stress levels to specific chromatophore cells to contract or expands and because of this there skin reflect light differently as contraction and expansion changes their pigment distribution ands changes colour accordingly.
Arctic Fox changes their coat which contains different pigment . Different pigments reflect light differently so colour change. For example white coat ( contain little pigment) in winter and brown coat in summer ( contain more pigment).
https://youtu.be/IgDGTLRcwa8
Squid have both photophores and Chromatophores within their skin. Because of this Squid changes their colour by controlling both the Colorado and the intensity of light produced.
In some squid Bioluminescence light may be produced by bacteria that live inside the animals Organs.
https://youtu.be/Ojb1pxcSr5E
Cuttlefish have different combinations of Chromatophores in their skin cells. An individual Chromatophores is surrounded by a circular muscle which can construct and expand.
By constructing and expanding of their muscle all the Chromatophores contain with different different pigments. Because of this Cuttlefish change their skin colour.

Some animal species like Nudibranchs ( small sea creatures) which changed their Color by changing their diet. Depending upon their coral diet thier body deposit the pigment from the coral in the skin and their Color change.
Some fishes change their Color according to environment of surrounding. They released different hormones which change the pigment . Because of changing pigment it's change colour.

Source Of Information: 
https://animals.howstuffworks.com/animal-facts/animal-camouflage2.htm
https://www.livescience.com/33159-chameleon-color-change-isnt-all-about-hiding.html
https://en.m.wikipedia.org/wiki/Animal_coloration