A counter-culture is a group that has values, beliefs, interests, or etcetera that conflict with the values (etc.) of the larger culture that that group is in.
These can be people who are gay, hippies, punk, goth, members of street gangs, pacifists, KKKers, or any of a hundred-thousand other interacting populations. Counter-culture is often used to imply that something is wrong with these people, but as a sociological term, it just means that there is some (important) aspect of their life-style that the larger culture disapproves of.
All counter-cultures are subcultures, but not all subcultures are countercultures. ('College students' are a subculture, but most would not consider them a counterculture)
Often 'counter-culture' will be used to refer specifically to the movement of youth in the 1960s away from traditional values, characterized by being anti-authoritarian, anti-war, pro-environment, pro-feminism, pro-drugs, the primary movers of the sexual revolution, and big on rock and roll. This is really only one of many counter-cultures.