It is a classification or description of a diverse spectrum of men who have consciously decided to avoid legally marrying, depending on a woman, or having a woman depend on them.
Men take this approach out of self defense; for preservation and protection. They see legal marriage or relationships in general in today's culture as too risky for men, laden with obligations and sacrifices with little to nothing to be gained by them. MGTOW seek to disengage or minimize involvement in systems and situations in which they think men are at a disadvantage.
Just about anyone who would consider himself as MGTOW would also not cohabitate with a woman. Most would avoid exclusive relationships entirely. And many avoid any ongoing interdependent interaction with woman. (This isn't to say such men necessarily shun their mothers, sisters, etc. This is mostly about romantic/social relationships and often professional relationships.)
It is too simplistic to say it is a reaction to feminism. Many MGTOW fully support equality for women under the law, in as much as it is possible. MGTOW a reaction to what is seen as misandry in our common systems and culture, especially our family laws and courts. In fact, some MGTOW could claim also to be feminists is that they support women being equal under the law, strong, and independent.
MGTOW isn't an organization. There are online forums, bloggers, and vloggers who identify as MGTOW, but there isn't a formal movement. Individualism is common among MGTOW. Men who identify as MGTOW might date or not, might be promiscuous or virgins, might have a religion or not, might be politically affiliated or not, might espouse an overall political philosophy or not, might believe in conspiracy theories or not. Some have previously been married, some are fathers. Some have never been through the wringer of family courts, but you don't have to be hit by a bus yourself to know you don't want to stand in the street.
There are no identifying qualities of a MGTOW man, aside from his self-identification as such, other than a rejection of current marriage laws and dominant cultural expectations for romantic relationships.