If you want to talk about men as a group (as in, "men like sports"), you would use "erkek/erkekler". If you want to talk about a specific man (as in, "that man over there"), or a specific group of men (as in, "those men over there"), you would use "adam".
Adam/Adamlar = Man/Men (as a specific person or specific group)
Erkekler = Men (as a general group or concept)
Erkek = Male
Bay = Mr (used in front of the last name, as in Bay Einstein)
Bey = Gentleman/Sir (used after the first name, as in Albert Bey)
Kadın/Kadınlar = Woman/Women (as a specific person or specific group)
Kadınlar = Women (as a general group or concept)
Bayan= Ms (used in front of the last name, as in Bayan Curie)
Hanım= Lady/Madam (used after the first name, as in Marie Hanım)
Note that Turkish does not have a word for "Mrs", so you cannot tell from a woman's name if she's married.
Erkek is used for both human man and male, there's no separation of the two, and if you asked a Turk for the "opposite" (so to speak) of kadın they'd say erkek. Confusingly, since erkek is used both for male things and men, if you asked the opposite of erkek you might get kadın (human woman), kız (human girl), or dişi (female).
Adam also means man most neutrally, but semantically it overlaps more with "insan" (person) or "kişi" (individual) rather than male.
Let your eye kind of wander over the different idioms using erkek and adam and I think you'll get an idea what I mean: