A man who is content with nontraditional gender roles; i.e., he is not threatened by intelligent and/or powerful women, and he does not have to be in control of every situation to maintain his sense of self. (Frequently, he does manifest a quiet kind of confidence and control over his surroundings, but it's not important to him that this is noticed by others.)
A beta male is often introverted, intelligent, and introspective. Though he may have been branded a "nerd" growing up, the adult beta is frequently a thoughtful, capable, and fascinating individual whom many women find appealing.
Alpha males, especially those of lesser intelligence and abilities, tend to feel threatened by beta males, who are increasingly preferred by women in modern times.
Peter Parker (Spider-Man) and Jim Halpert (The Office) are famous examples of beta males.