What kind of factors determine how someone looks according to their race or ethnicity?
Concept of race

The concept of race often poorly represent the genetic structure of populations. There is therefore an important socio-cultural aspect to the concept of race. The first sentences from wikipedia > race make this point clear


  Race is a concept used in the categorization of humans into groups, called races or racial groups, based on combinations of shared physical traits, ancestry, genetics, and social or cultural traits. Although such groupings lack a firm basis in modern biology, they continue to have a strong influence over contemporary social relations.


What explain why people of different race look different?

You should most definitely have a look at the post Why is a heritability coefficient not an index of how “genetic” something is?

Your question can therefore be rephrased as "What is the heritability of the phenotypic trait X in humans?", where trait X could be anything of interest such as hair colour for example.

The answer for such question is always the same. It goes in three points


It depends upon which phenotypic trait you are looking at
For whichever trait you consider, it will most certainly be a combination of both genetic and environmental factor.
The evolutionary reasons for the existence of such distinction among groups will again depend upon which trait you're looking at.


If there is one particular trait that is of interest to you, you might want to open a new post to ask what is the heritability of this trait and whether there is any selective reasons for why different ethnic group differ for this trait.



You might be interested in a short and easy intro course to evolutionary biology such as Evo101 by UC Berkeley