This will probably be a short one.

So there are a lot of mysteries and misdirection offered about the dark web, today I'm going to shed as much light as I can on the subject.

The onion router(Tor) network is a series of volunteer run nodes(or computers) that allows people to route their browser traffic through them to avoid things like government surveillance and censorship. How it works is that your method of accessing the TOR network(which I recommend only using the Tor browser for) will select three of these nodes, an entry node, a relay node, and an exit node. Each one of these serves a crucial function. First your browser uses the public information from each of these relays to encrypt your data requests, first the exit node, then the relay, then the entrance node. The entry node is where your requests enter the Tor network, your browser sends the encrypted information to the entrance node, which is able to see your computer's IP address, but not that of the site you are visiting. This connection is similar to a VPN, but I will talk about the differences later. The entrance node then removes the first layer of encryption and sends the data to the relay node. The relay node does not know your computer's IP address, or what website you are visiting. The relay node removes the second level of encryption, and forwards the information to the exit node. The exit node removes the final layer of encryption, and with that is able to tell which website you wish to visit, and sends out your request to the open web, however neither the exit node, or the website you visit can see your IP address, or your location.

Theoretically this provides near perfect anonymity to anyone who wishes to use the Tor network. Indeed the NSA(People say that the FBI is spying on American's internet usage, it's actually the NSA, but for all intents and purposes you can consider them the same), has admitted that mass surveillance over the Tor network is impossible. Believe me, they've tried. However, it's not fool proof, and there are some weaknesses that we need to be aware of.

1. The Tor network isn't a free VPN. Now if you really just need to hide from your ISP or a network administrator such as your school or work, it would probably work just fine. They can tell you are using the Tor browser, but nothing else. However you can't choose the exit relay, and that does come with some vulnerabilities, such as not having control for geo-restricted content, and some relays have been found downgrading HTTPS connections to HTTP connections. Do not use HTTP connections over the Tor Network. Honestly you probably shouldn't use HTTP connections for any reason, but especially not over the Tor browser.
2. If someone such as the NSA has the ability to monitor nearly all internet traffic they can over time build profiles. I.e. you started using the Tor network at this time, someone downloaded 45GB of content off of this website via the Tor network five minutes later, you received 45GB of data at the same time, and then both you and the exit relay that made the download turned off your computers at near the same time. Now they can't necessarily prove it was you from just that, but they can build a profile from it.
3. If someone controls the entrance, relay, and exit nodes they can see who you are and what websites you visit. Multiple government organizations run Tor nodes for that reason.

The dark web is similar, only that not only are you using the Tor network to connect to the website, but the website is using the Tor network to connect to you, meaning that a total of 6 nodes are being used, hiding both of you from each other. URL's on the dark web are completely randomly generated, but they all end in .onion. So you won't see something like 17thshard.com, instead you would see something like Qwfidanl54/dhffeihfjs8fhsih9rh=fshi.onion. That's not a real URL, don't try it.

A lot of propaganda says that the dark web is only for criminals or hackers, that's not even close to true. While some do use the dark web, the dark web was created for a much higher purpose. Journalists and ordinary citizens in authoritarian countries use the dark web to find or post news the government wants to suppress, or simply to access the free internet. Researchers will post their findings there. The military uses the dark web for all kinds of things. Even businesses will sometimes use the dark web to create anonymous tip lines or to allow whistleblowers to report on their bosses without threat of punishment, even Facebook has a dark web domain.

In short the Tor network and the dark web are useful tools for the world, and greater understanding is needed in regards both to what they are and to what they are not.

A few other things really quick

If you are worried about your internet service provider(ISP) or local government flagging you as a Tor user(as some do), you can use a VPN to connect to a server in a freer country and access the Tor network from there.

As access to the Tor network is more difficult in some countries due to government crackdowns the Tor project has created snowflake to allow volunteers in free countries to operate bridges that will allow people in places like China, Iran, or Egypt to have free access to the Internet, by making it look like they are having a video call with you, https://support.torproject.org/anti-censorship/what-is-snowflake/.

Which is something I plan to do once I have the resources to make it a reality.