The wage gap is a real thing isnt it?
No. The wage gap is what you get when you compare the average income of all men, versus the average income of all women, and you find out that collectively, women make less than men accounting for a 23 cent difference. The reasons why include the fact that women take more time off, retire earlier, seek jobs that are lesser pay, and are less likely to demand raises than men are. On an individual same-job comparison, though, women and men make exactly the same. If employers could get away with hiring women for less money, it would actually be a positive for women, and men would have trouble with unemployment.
As for what feminism is, the term originated in the late 1800s, but feminist writing could be found a century earlier in private writings. Women back stridently demanded the right to vote, own property, dress how they want, hold public office, and so on. The feminist movement during this time faded out because all the demands were too much of a culture shock, so nothing of political meaning really took place.
Fast forward to the 20s, and you have the suffragist movement. Women like Susan B Anthony and Alice Paul knew the history from before, and decided that even though women faced a lot of inequalities, they would settle for tackling the biggest issue:Voting. If women could get a political voice, then the rest of their rights would come later. Through public protests, including hunger strikes and civil disobedience, the 19th amendment would be passed.
Now we get to the 60s. Women have their right to vote now, but a different kind of battle is taking place. Women at this point are still mostly confined to the home, and expected to be housewives. Betty Friedan writes "The feminist mystique" to address the fact that women need more than motherhood to be happy, they also need a trade or a career to give themselves a unique meaning. This kind of change would be more gradual, and would become visible in the 90s, with women graduating college at a higher rate than men in the 2000s.
Why there are still feminists today is a bit of a tougher question to answer. Legally, there are no restrictions in life based on being a woman, and women can own property, divorce at will, and get any job they want within reason. There really are not any more battles to fight, but modern feminists seem to have missed the memo. Feminism, in a period of around 80 years, rapidly transformed a woman's place in the world. It could be that there are still feminists today because the change happened so quickly that a lot of people haven't even adjusted to the change, and are still used to the old way society was.
It could also be that a lot of political lobbying in the interest of women would become irrelevant if people didn't think feminism was needed in the 21st century, so a lot of false narratives about wages and rape have to be propagated in order for these organizations to stay funded and relevant. A similar phenomenon currently faces the black rights movement, with people like Al Sharpton who blatantly profit off of social divides. How long stuff like this can go on, I am not sure, I hope it will end sooner than later.