Although the concept of socialism did not start with Marx, it is helpful to remember his analysis of capitalism, to which socialism can perhaps be better understood of as a response. According to Marx, capital is accumulated labor: capitalists, those who own the means of production, become wealthy by keeping for themselves some of the value created by their employees. Capitalists have a vested interest in extracting as much value from their workers as they can while compensating them as little as possible, producing in the process an ever-growing gap in the bargaining power between the two camps (just look at the gap between rich and poor in the US). According to Marx, labor—which under natural conditions is a source of meaning, value and identity—becomes under capitalism the greatest source of human alienation and exploitation.
Socialism, then, can be understood as an attempt to check the imbalance and exploitation inherent in free-market capitalism. In its worst and most perverted manifestations, however, 'nationalist socialist' regimes have tyrannically appropriated the means of production and taken on the role of the capitalists without protecting the interests of the people, predictably leading to the social and/or material ruin of their countries. Such approaches have merely substituted one oppressor and form of corruption and greed for another while illegitimately maintaining the title of 'socialists.' Such tyrannical and corrupt systems are usually referred to as Marxist (or sometimes also Leninist) Socialism. As an indefatigable defender of freedom, however, Marx became so appalled by the misinterpretation and the misapplication of his ideas that, according to his life-long friend and intellectual collaborator Friedrich Engels, he once declared "All I know is that I am not a Marxist."
In its best manifestations, however, most notably in democratic socialism, where the government is beholden to the people, wealth is made to work for the protection and welfare of all members of society: the incentives for economic success and upward mobility are there, but opportunities for success do not depend on the accident of inherited socio-economic status. Rather, wealth is used to create the conditions that make it possible to meaningfully empower everyone to be able to adequately pursue their own individual conception of the good, and then to pay it forward. Under democratic socialism, these material and structural conditions—quality education, healthcare, unemployment protection, safe working conditions, housing, regulations for economic and environmental sustainability, etc.—are understood as basic human rights, owed to every single citizen, not as the privilege of a wealthy elite.
To most Americans this may sound like an unrealistic pipe dream. But the proof of concept already exists, most notably in the Nordic European countries (Sweden, Finland, Norway, Denmark), as well as Germany, Australia and New Zealand, all of which consistently score highest on the various indices used to measure well-being: education, health, safety, civic engagement, gender equality, environmental sustainability, life satisfaction, income equality, work-life balance, working conditions, parental leave, rates of recidivism, etc. These remarkable rates of success can be at least partially attributed to the importance such governments place on enacting policies that are supported by comprehensive social science research, and to their strong commitment to protecting individual rights. This is not to say that these countries don't have problems of their own, of course, but they do represent an alternative model to the good life that is well worth considering, and possibly emulating...
If socialism is such a great idea, why hasn't it taken root in the United States? Well, ignoring the Pavlovian mental associations forged in the mind of Americans, especially during the Cold War, and given our narrow conception of competitive, boot-strapping individualism and the American (pipe) dream of upward mobility, there's a famous quote, apocryphally attributed to John Steinbeck, well worth considering, which argues that:
Socialism never took root in America because the poor see themselves not as an exploited proletariat, but as temporarily embarrassed millionaires.
Historians, ethnographers and sociologists know what I'm talking about...