It has to do with the different reflective properties of their retinas.
Many nocturnal animals have a mirror-like layer at the back of their eyes called the tapetum which helps them see better in the dark. You can read about the tapetum here
http://www.pbs.org/wgbh/nova/leopards/nocturnaleye.html
The red reflection is due to the haemoglobin in the blood in the retinal area and the lack of pigmente there.
When tapetal pigment is present, its color dominates the color of a given animal's reflection and this color somewhat loosely follows coat color in many animals. Green being seen more in darker animals. Different types of animals can have different pigments or absent pigment resulting in the eye shine you see.
Hope this helps you!

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