If i have a brown eye gene which encodes the protein that is responsible for the brown color, and have a blue eye gene either, what is the reason that my eye color is brown? How does one gene maintains dominance to another one?
One version or allele of the gene for eye color may encode instructions to make the brown pigment, while the blue allele makes no pigment. Alternatively, it is possible, logically speaking, that the blue allele blocks the productions of brown pigment, but this is not the case.
Now, brown vs blue is not a case of one gene and two alleles. After producing pigment, it must be transported, deposited and such. Many processes are involved. So, the pigment deposition process may be functional or not, but as above, it is often found that a non-functional version is simply non-functioning and not actively inhibiting the process. There are numerous cases where an aberrant form of a protein, encoded by a variant allele, acts as an active inhibitor of a given biological or biochemical process.
The key to your question as written is blue eye color is a lack of pigmentation.
At the DNA level there are several mechanisms that might be cited.
A mutation (change in the DNA sequence) of a gene may actually render one copy of the gene defective. In the case of albino-ism, the skin is pink - no melanin pigment is made at all. This is a case of both copies of a vital gene being defective and an ability lost completely.
Some cases of genetically inherited diabetes (<1%) dwarfism and obesity are examples of this. There are many others - most genetic diseases are like this.
This can be caused by a single mutation in each defective gene, and others an entire segment of the chromosome can be lost or disrupted by a new, often nonsensical sequence dropped into the gene region.
Other changes can affect the way the genes turn on and off. - similar sequence changes in the regulatory sequences next to the gene sequence will make the gene behave differently. Even babies of african descent may be born with blonde hair and blue eyes and get the darker pigmentation later. this is genetically caused by a change in they timing of the genes for hair and eye color turning on at other times (blonde hair and blue eyes have little or no pigment, so if the genes are off, then this is what you get). So when genes are active can cause dominant (always on) versus recessive (sometimes off) patterns. which are because one copy of the gene behaves differently than the other copy.
Sometimes a single mutation in one copy of the gene can cause rather interesting effects. A single good gene and a mutated, less effective gene might result in lost of activity because the mixture of the two genes is different between a wholly functional or wholly variant set of genes.
A classic example is the sickle cell trait of hemoglobin. two copies of sickle cell hemoglobin causes the red blood cells to be stiff and misshapen which in turn causes a painful condition in the carrier. a person with a single copy of the sickle cell gene does not have this problem - is nearly asymptomatic. So we say sickle cell anemia is recessive for the HbS mutation. but in reality there is plenty of the HbS version of hemoglobin (Hb) in the blood cells, just that the mixture of the two does not cause the anemia.
There are lots of other specific cases on how dominant and recessive effects happen, but this might give you a picture.
Sorry to blockquote, but this note from wikipedia gets to the heart of the question:
Recommended reading for an overview:
You can also look to "dominant negative" phenotypes to get a good idea of what "dominance" entails (this was the concept that really locked it in for me in college). One example is Marfan syndrome, where the mutant allele of FBN1 (fibrillin-1) produces a version of the protein that is antagonistic to the protein produced by the "healthy" allele.
Dominant negative and semi-dominant alleles can be seen when the protein forms dimers to function. Activin type receptors form dimers to transduce a signal from an extracellular ligand into the cell. A truncated receptor protein that has the dimerization domain but lacks the intracellular domain necessary for signal transduction will act in a dominant negative fashion over the wildtype allele: Dimers are formed but the signal is not transduced. This has been implicated in hereditary colorectal cancers.