Aether is an app you install to your computer to connect to Aether network. This network is made of different boards (forums) where people post and discuss things. On the surface, it's fairly similar to Slashdot, Metafilter, Reddit, or any other community site on the Internet.
The different thing about Aether is that it doesn't have a server somewhere. The only thing the app does is that it finds and connects to other people using Aether. In other words, it's a distributed, peer-to-peer network.
This makes it impossible to censor, and renders its users anonymous. It's useful for people concerned about privacy, or pretty much anyone who doesn't want to be watched and catalogued for every word they write on the Internet (so, pretty much everybody).
It's also temporary. Whatever you post disappears after six months. It's designed to be an ephemeral space, and it's focused on now, rather than the past. Other people can still keep copies of what you wrote, but it won't last forever in the network itself. They also won't know who you are.
I believe in free speech.
Aether fills a hole in the fabric of the Internet; it's the thing you use to talk to other people when everything else is down or inaccessible. If the entire web died tomorrow, all the web servers blocked, Aether, and its boards would still be alive and thriving. It might not happen in your country, but it happens at a country every day.
It's fairly resilient, nearly impossible to block (since there is nothing to block), and useful in cases where dissemination of information by other means is not guaranteed or outright prevented.
Aether doesn't depend on the web. This is possible because it's nothing but a communication protocol that describes how you find other people speaking Aether protocol, and how to speak to them. Aether (the app) is just a reference implementation of a client that talks Aether protocol. The protocol itself is open.
It's also an experiment. Don't trust it just yet. But definitely give it a shot!