Two different mechanism exist which determine the color of an object we perceive.
The first is controlled by pigments like melanin as mentioned in other comments. Pigments absorb certain wavelengths of light and reflect others. We then see the reflected light. Our skin gets darker when a higher pigment concentration absorbs more light.
The second mechanism is light scattering. Think of it as physical nano structures that bend light into other wavelengths. Blue pigment is very rare in animals and most get their blue color this way. Some fish and amphibians are the exception.
The human iris gets it’s color via a combination of both.
A bunch of genes influence the pigment production. If you get a high yielding set form at least one parent, most of the light is absorbed by pigments and you’ll have brown eyes. Barely any light is left to scatter.
Blue and grey eyes are where you produce less pigment so only a bit of the light is absorbed and the rest scattered. Grey is rarer because it requires more specific scattering.
Green and hazel eyes lie in between. Green is so rare because it needs just the right amount of pigmentation.
Amber may look like hazel but is produced by a specific combination of pigment concentrations and less a result of scattering. Hence it’s the third rarest after grey and green.
No pigments at all means the light goes straight through to be reflected by blood vessels causing the red eyes characteristic for albinism.
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