According to dictionary.com, a counter-culture describes a lifestyle, usually pursued by young people, that rejects the values and norms of the dominant society. The term, it notes, was first coined in the 1960s.
At that time, a youth movement was developing in response to the increasingly unpopular war in Viet Nam. Some youth questioned why the United States was at war halfway around the globe, and why they were being drafted to fight in it. This led to a wider questioning of how United States society was structured, who had power, and why the country seemed so fixated on material goods and consumption. Counter-culture youth grew their hair long (crew cuts were associated with the military) and often rejected the idea of a nine-to-five office job with a house in the suburbs in favor of living on the margins of society, free to pursue their own interests. Counter-cultural youth wore old clothes (such as jeans and bleach-stained teeshirts that soon became the stylish "tie-die"), and often went barefoot. They grouped themselves into communes and shared housing, or lived in parks to keep expenses down. This movement dovetailed with the Civil Rights movement, in which questions were raised about a social order that oppressed minorities, and out of this questioning the women's movement also came to the forefront with renewed vigor. Youth movements spread to Europe as well, leading to protests similar to those in the United States in opposition to the Viet Nam War, and questioning the relevance of university curricula.
Society definitely changed as a result of the counter-culture movement. Women gained more legal rights against discrimination and more control over their bodies, as well as more social acceptance as they expanded their roles in the workplace. The voting age was lowered from twenty-one to eighteen, and the draft ended. Dress codes relaxed, and society as a whole became more open to discussing formerly taboo subjects, such as sex. Censorship codes were lifted from movies. American culture (and others) underwent a vast upheaval, the results of which still live with us today. Some segments of society continue to embrace the greater tolerance and social leveling the counter-culture movement ushered in, and others segments of the population would like to take the country back to what seems a more ordered and traditional way of life.
Some people still live in the counter-culture, and some are turning to it. There are many different cultures now, and it's hard to define exactly what is not normal. With the proliferation of social media, people can try new social groups and new ideas very easily. Some of them will stay, and some will just sample and move on. But the groups and their ideas are easier to find.
The long term result of the counterculture is the sort of cultural conflicts that we have today between Americans who hold to more traditional values and those who believe in more liberal/progressive values.
In the '60s, America started to be torn between the old values (religious faith, marriage, patriotism) and a new set of values that were more secular, individualistic and skeptical of things like religion and patriotism. This was manifested in such things as the hippie movement, the sexual revolution, and the Women's Lib movement.
We continue to see debates between these sides today. Conservatives and liberals continue to fight over issues of morality like abortion and gay rights. There continues to be debate over the proper role of women in society and over the degree to which the Christian faith should play a prominent role in public life.
In these ways, the debates that started with the rise of the counterculture continue to be with us today. This is the major long-term result of the counterculture.