How is one allele dominant over the other?

One trait's dominance just means that it gets expressed, whereas the recessive trait does not. Usually, a dominant allele codes for a particular enzyme that a recessive allele does not. As long as a dominant allele exists, that allele's product will exist, so it masks the effect of the recessive allele; the recessive allele only gets expressed when there is no dominant allele present.

The recessive trait only gets expressed when identical allels are inherited from each parent.  The dominant trait requires only on alleel for expression.  

If your asking what determines the fact that a trait requires either one allele or two identical alleles for expression, I don't think that is known.

Just to point out, not ALL genes are completely dominant, there are a whole range of different types, but we always start with the complete domance and recessive alleles first and its less confusing.

In complete dominance you will get a trait or not. ie there are 2 possible options for phenotype...the dominant one and the recessive one (which of course needs 2 alleles to be expressed).
A lot of people would think that traits are more like adding red and white together to get pink...a mixture or intermediate features as this makes more sense (this is actually called incomplete dominance and gives you offspring that can be red, pink or white)...some other genes will show BOTH alleles eg blood type where A and B are both dominant, while O is recessive (this is called co-dominance).

I know sometimes I also wonder how can you have 1 form completely dominant over the other and not even see the trait a little bit.....I guess maybe you have to think on a more biochemical level where the presence of a allele, if present will cause a biochemical reaction to occur....if its not there, this process does not take place. Eye colour (ok this has modifier genes as well but ignore that) where is the brown allele is present it caused a pigment to produce brown if its there...if no allele for pigmentation is there, you get blue eyes...so maybe think of recessive things as a LACK of pigment, a chemical, or something...When its not there you show recessive, but even if its "half there" eg Aa, the biochemical process occurs.

Long story I guess to try and explain it, but I just wanted to point out that there are other types of inheritance as well as the ones you are learning about....these tend to give you a range of things like height, skin colour and things like intermeidates which you would tend to expect...... I guess it goes back to Mendel as everyone did expect things to be passed on in mixtures, but he found that was not always the case with his peas....if it wasn't for complete dominance we may still be in the dark about inheritance.