What makes someone attractive psychologically? What makes a person particularly attractive to another person, specifically the opposite sex? 
Why would the brain evolve this trait to target specific genes (that make someone look pretty), and exclude others (that make someone look ugly)?


This seems to differ from the accepted idea that ancient female humans were attracted to stronger, more able bodied men due to survival instinct. Facial features seem to have nothing to do with survival or natural selection.


So, why are facial features significant to assessing beauty?
Also, how does the brain process physical features as good or bad?
With regard to your first question about the psychological processes of interpersonal attraction there are (at least) 4 factors that have been found in the social psychology literature.


Contextual Aspects. 
People are more likely to develop attraction
towards those they see more frequently than others. This is known as
the Mere Exposure Effect (Saegert, Swap &amp; Zajonc, 1973). 
Attributes of the Target Person. 
The perceived attractiveness (for example of the
face) is another factor.  The comments point to resources regarding
this aspect. 
Attributes of the Perceiver. 
Mood has an effect on interpersonal attraction. Being in a positive mood makes a person more likely to develop attraction (Gouaux, 1971).  
Aspects of the relationship between Perceiver and Target Person. 
Perceived similarity has an effect on interpersonal attraction (Byrne, 1971).  Similarities can serve as a positive reinforcement of the attitudes of the perceiver.
They are also the foundation for common activities and also which in
turn lead to closer relationship.


References

Byrne, D. E. (1971). The attraction paradigm (Bd. 11). New York: Academic Press.

Condon, J. W., &amp; Crano, W. D. (1988). Inferred evaluation and the relation between attitude similarity and interpersonal attraction. Journal of Personality and Social Psychology, 54(5), 789.

Gouaux, C. (1971). Induced affective states and interpersonal attraction. Journal of Personality and Social Psychology, 20, 37–43.

Saegert, S., Swap, W., &amp; Zajonc, R. B. (1973). Exposure, context, and interpersonal attraction. Journal of Personality and Social Psychology, 25(2), 234-242.