How is possible my sister has green eyes?

All that gene stuff is nonsense.  She has green eyes because she is an individual and everyone is different.

With a brown-eye parent it is possible to get a child with any colour, only the most probably colour is brown at 55%. It's still possible to get a child with green or blue eyes.

Your father could carry a gene for green eyes, but eye color is not as simple as just two genes. Genetics is complicated. I'm sure if you look back in your family someone has had green eyes. Don't let it bother you. People's eye color can change over their lifetime as well, it's just a matter of the pigment their body is producing. Don't worry about it.

green eyes are part of the brown-eyed gene

Eye color is a polygenic trait. That means that many genes affect the color of the eyes, not just one gene as some traits are. Because of this, there are so many different shades of eye color, not just your typical brown, black, blue, and green.

This seems like an example of codominance. Codominance is when both alleles (factors determining a characteristic) contribute to the phenotype (what you look like).

Eye colour is not as simple as one gene, 2 alleles , 2 colours.

In fact eye colour inheritance is not really totally understood. Some people think there could a number of different genes involved, up to 6 and the different combinations of these alleles give you the variety of eye colours you get. eg AaBBCcddEe.

There is also other genes involved called modifier genes which can alter the original colour.

My mother had light blue eyes and my father has hazel eyes with large yellow and green flecks.
I have green/blue eyes - I have a yellow circle at the center over the blue base which makes them more green.

My sister has got dark blue eyes. one daughter has light blue eyes (the other gets her hazel eyes from her her father)

My brother has got more hazel, but he calls them blue. They are in fact a dark blue base with a hazel overlay, so they appear either gray or hazel. His daughter has eyes like him - more gray.

So none of us have eyes like either parent.

So your sister probably has some genes from your brown eyed father (who could have genes which have green colouring but you can't see it because it is masked by the brown) and there could be modifier genes which alter the colour. I think that is what happened to me because as a child I had blue eyes but they changed when I hit my teens.

Heritable traits are a function of both reading a gene (producing protein or RNA), and the environmental variables present while the gene product is acting. In the case of eyes, part of the final color comes from gene product, and part of it comes from how fast and how long this gene product migrates through tissues, as well as the properties of those tissues. We're all just a little bit different genetically and we all develop in environments that are a little bit different, so our various phenotypes will never be entirely predictable.