Reddit is essentially a message board with a voting system that decides which topics and comments are most visible. Topics can be text posts, links, or images. It is divided into subject-oriented "subreddits" that are created and managed by volunteer moderators. There are a set of default subreddits that appear on reddit.com if you don't have an account, and which you are subscribed to by default when you create an account. By creating an account, you can unsubscribe to the defaults and subscribe to any of the thousands of other subreddits that exist.
Subreddits range from amazing, to inane, to noxious. The good include /r/personalfinance, which offers great financial advice; /r/askscience, which offers expert responses to interesting scientific questions; and /r/iama, which regularly hosts celebrity Q&A sessions, most famously once with President Obama. A more niche one that I enjoy is /r/colorizedhistory, which colorizes historical photographs. Other subreddits are arguably the best place on the internet to discuss the particular topic at issue--true for many video game and sports oriented subreddits (e.g. /r/mma and /r/magictcg). I myself got involved with reddit because the developers of Minecraft treated /r/minecraft as essentially their official forums.
The inane unfortunately includes a fair bit of what you find in the default subreddits, and contributes to people thinking that reddit is dumb. Stuff like /r/showerthoughts, /r/adviceanimals, /r/aww, and /r/til. But hey, maybe you like inane stuff to read on the toilet. Reddit is good for that too if that's what you want; you just need to subscribe to what interests you.
There are also tiny subreddits devoted to hateful things. Many are just created by trolls to be offensive on purpose to rile people up. Some are genuine hateful communities. These are largely invisible to redditors themselves, but get outsize attention from media with an ax to grind who seek them out in order to "expose" reddit.
As far as recent controversy goes, there are essentially two main issues. First, reddit was historically set up as, in the words of one of the founds, a "bastion of free speech." The plan was to censor only illegal things to allow people to freely discuss ideas. That's why the noxious subreddits were allowed. Recently, the reddit admins have been changing that policy and want to start policing the community more heavily. Some users are happy about this because they want the noxious communities gone. Some users are unhappy because they are part of the noxious communities themselves. The majority of redditors, I would guess, are no fans of the noxious communities but are fine to let them exist because they worry about where the admins will draw the line once they are free to censor things.
The second issue is the lack of support for moderators. Reddit's moderation tools are apparently atrociously bad, and Reddit's paid administrators communicate poorly with the unpaid moderators. This came to a head recently when Reddit fired the one administrator who did provide moderators with a lot of help, causing temporary chaos for a few major subreddits who relied on her. In protest, many huge subreddits shut down for a couple of days. Reddit responded by promising improved tools by a certain date. Bust just yesterday, Reddit's chief engineer just quit because she said that Reddit can't deliver on that promise.
The drama aspect comes in with Reddit's temporary CEO, Ellen Pao. She was widely blamed for the issues above by users, and stepped down this week. But now information is coming to light that the board, particular one of the founders, was really responsible. So people who care about making drama are turning that all into a big thing.