In short: women want dominant men as short term mates and prestigious men as long term mates. Dominant males being those who are ambitious, assertive and boss other people around, intimidating others. The kind of alpha males chasing desired women. The prestigious kind are those who achieve high social status through social networks (twitter?), making use of useful social aptitude and abilities or good social skills such as being able to listen and make eye contact, giving compliments.
The main purpose for this is efficacious reproductive strategies, to keep mankind on top of the evolution. It was even thought that high-dominant men had high testosterone levels. Men with high levels of testosterone are more likely to engage in short-term mating behaviors and less likely to engage in parenting. But the relationship high dominance and high testosterone could not be found.
The other way around, thus high prestige men having low testosterone levels isn’t studied yet. Nevertheless, low testosterone levels are associated with a decreased likelihood of physical aggression and intrasexual competition. Low testosterone is associated with fatherhood and an increased likelihood of offspring care.