Here is a very short but accurate summary for you. Feel free to repost it.
Some guy broke up with a girl and posted some unverifiable and irrelevant stuff about her on his blog. This started a campaign against her, with increasingly outrageous lies. It was said she slept with a journalist in exchange for a favourable review of her game, but the review doesn't exist and the journalist in question didn't write anything about her after they got together. The GamerGate keeps trying to claim it is interested in journalistic integrity, while repeating this lie over and over again.
Another women, Anita Sarkeesian, made some videos about how games tend to have fairly poor portrayals of women. She was careful to point out that most of it was due to laziness on the part of the developers, even calling the series Tropes vs. Women to emphasize the fact. Even so, this sparked a campaign against her by self-described "gamers" who thought she was attacking games and gaming culture. GamerGate supporters have tried to distance themselves from some of the worst of it (doxing, rape and murder threats, and recently a bomb threat against a university she was due to speak at), while posting conspiracy theories about she sent all these threats to herself... in order to lose money by not being paid to speak or something.
Other female developers who dared to speak out have received similar treatment, such as Brianna Wu. GameGate supporters have also organized somewhat successful campaigns to get advertisers to stop supporting web sites which condemned the treatment of these women. The label "gamer" has come to mean a misogynistic, unpleasant fanboy who masturbates over pixelated tits and nude patches, and should be abandoned by normal people who like games, half of whom are women.