It all depends on the particular mutation (not gene, but allele would be an acceptable term as well). Thus, we'd have to look at it on a case by case basis, though we can make a few generalizations.
Let's start with your example. Brown eyes is dominant and blue eyes are recessive.
We have to ask "what makes an eye brown instead of blue"
Knowing this, it may now be clear why brown is dominant. The gene that results in the brown pigment is either present (resulting in black/brown) or a mutation has occurred to disrupt this gene, resulting in no pigment production, and thus blue eyes.
With many alleles, this is the type of scenario that plays out. The dominant allele is simply a functional gene that results in a product. The recessive is frequently a mutation that abolish production of that product. Thus, with simple mendelian genetics, if you have a single copy of the domiant gene, you are good to go (as you have at least 1 functional allele, thus the gene product is made and does it's job). You need both of the recessive alleles to get a phenotype, as the phenotype results from complete lack of a function, thus you need both copies of the gene to be non-functional (which is basically the definition of recessive).
It can get much more complicated than this, AND there is much more than the simple mendelian dominant/recessive. For example, you have dominant negative (Huntington's), incomplete dominance, co-dominance (A/B blood types), more than 2 alleles (ABO blood types), and then there are traits that are influenced by a large number of genes (height).Source