As I'm sure you know, it's mostly recognized as a sex hormone in men. It has other functions as well. It regulates fat distribution, muscle mass, bone density, aids in the production of red blood cells and sperm, there may be some function with heart health but its unclear as of yet what has might be.
Mentally and behaviorally, there's been a correlation between low testosterone and poor self esteem, sleep problems, depression, lack of focus, lack of energy, etc. Also, some conventional wisdom stemming from past correlation studies regarding competitiveness and aggression has recently been debunked so we'll see how that play out in time.
Where are you going with this?