Let's say you want to go to reddit.com. You type it into your browser and suddenly you're there. But this actually happens in a couple of hops. First, you need to find the .com server. Once you've found that, it will tell you where to find reddit.com. Once you're at reddit.com, it'll tell you how to find r/spacedicks or wherever you want to go.
But hold on a minute - who tells you how to get to .com in the first place? That would be one of the 13 root servers run by different organisations around the world - some are run by the US Government, and others are run by private and/or foreign organisations.
The list of all of the top level domains, like .com, is maintained and published by ICANN in its root file. If you wanted to create a new TLD, like .explanation, you'd need to get ICANN's approval. Once you have that, it will be published in the root zone, and whoever controls .explanation can start selling/using websites under that domain.
But who is ICANN? It's not some faceless entity - it's like a big tent that everyone meets under to talk about the Internet. And they've got a really cool way of making decisions - consensus. Consensus is better than voting, because with voting you can have 60% of people screwing over the other 40%. With consensus, everyone has to agree. So everyone meets up at ICANN to make decisions about these kinds of things - which domains to let into the root (and other things). These people include private businesses, civil society, governments, academics, technical community, and others.
This group of people, called "the multistakeholder community" has been running the Internet for the past 20 years. They've done a pretty good job, that's why today it works so well.
When the US government created ICANN, they always planned to let the multistakeholder community take control of this. But because it was all new, they kept a special role for the government. They had a kind of "veto" power over ICANN - that they never really used - but the threat that they would use it kept ICANN honest. The US was like a benevolant steward, keeping an eye on ICANN and the community, and making sure it didn't screw things up. It was always the understanding that this was a temporary role, but because everything was fine, no one really pressured the US to step back and they were fine with the status quo.
Then Snowden happened and the world was really angry with the US. Some countries felt concerned that the US government had this special power. They started making noise and said that control of ICANN should move to the United Nations. This is a very bad idea - because when governments start making technical decisions for political reasons, you'll end up with crappy outcomes.
So the US Government annonced that it would step back from this role and transition oversight of ICANN to the multistakeholder community.
The Republicans have been making political hay out of this, because on the surface, it looks like out of nowhere, Obama's "giving away the Internet" and you can't explain why this isn't true in a soundbite. To believe this is to fundamentally misunderstand how the Internet works. This power that the US has isn't really a power at all. If the US had used it, say, to remove .ir from the root and knock Iran of the Internet, there's no way other countries would have tolerated this. They would have created a new ICANN and moved everything over. And there's no technical reason why they can't do this at any time. The only reason people stay with ICANN is because it works.
Now, you might say - "but now Iran and China will be able to knock .com off the Internet and harm/censor America's Internet!" But remember that consensus thing we were talking about? In the multistakeholder model, countries participate on the same level as all the other participants - academics, companies, technical people. So if China and Iran want .com to be taken off the root, then they'll need to convince the US, UK, France, Netherlands etc and also Google, Microsoft, Facebook etc. And if they can do that, then maybe they have a point!
What about censorship on a lower level? What if someone wants to censor reddit.com? That also won't work - because ICANN only operates at the root - so (conceivably) they could take down .com - but they couldn't take down specific domains under .com. Censorship happens on the country/ISP level - not at the ICANN level. Countries that censor their Internet will still do so after this transition, there is nothing that ICANN can do to stop that.
For the average Internet user, there is nothing to worry about. This is a minor, mostly symbolic change that the whole technical community supports - here you can see a long list of statements from various companies, industry bodies and civil society groups in support of this change - note the names of the companies/industry bodies that support this: https://www.icann.org/en/system/file...13sep16-en.pdf
Here's a brief video by the guy who invented the Internet explaining this:
Sorry for the long explanation, but that's about as simple as I can give it. If you look at my previous comments, you'll see I've been going across reddit trying to add some common sense into this issue, because some people are getting really bent out of shape about this, when they really shouldn't. I don't work for ICANN, but I do know a lot about this issue, because I've been involved (on a very, very low level).
TLDR: The world is not going crazy, the government is not about to censor your Internet.