Maybe you can help me understand a simple example - if there is a gene that controls the eye color, how does it express itself as eye color (vs something completely different)
It's important that there isn't a gene that controls the eye colour.
Eye colour is modulated by several genes which haven't necessarily been identified - I understand this to be an open question.
I can give an example, though:
Eye colour is a function of what pigments are present in the iris, the most relevant pigment (in humans) being melanin, which also contributes to dark skin and tanning.
So, melanin is produced via the biochemical pathway 'melanogenesis'.
Among other things, the process of melanogenesis requires the enzyme tyrosinase, a protein which has a decent length wikipedia page.
If you are missing the TYR gene which codes for the protein tyrosinase, then you will likely be albino. This won't only affect your eyes, then.
I say you will 'likely' be albino only because it's hypothetically possible that a lack of tyrosinase could be compensated for with another biochemical pathway. And, of course, albinism could come about by some other break in melanin production.
A gene is like a DLL. Asking how it works would require a lot of know-how that’s specific to the usage.
When talking about how DNA is an instruction for making a protein, that protein is your data structure or a function. The next level of fundamental building block. Genes can contain a lot of information or non-information for proteins. Just like a program might have a lot of functions or data. The way they work together and interact is what creates a larger effect.
I think maybe the best starting point for understanding biology is learning about enzymes up close. They are proteins that act as “machines”. It gets down to physics and chemistry to determine how they work, so that’s probably when to drop the analogies.
Basically, the variance in melanin is what causes different eye colors. So, if you have a gene that produces melanin, and then turn it's expression down, then you might get bluer eyes.
The complicated part is that a lot of genes actually control for eye color - that is where the complications come. Each gene interacts with every other gene in a unique way.