The threat of government retaliation for online speech hasn't been more clear in decades as it is now in the U.S. under the Trump administration. The Trump administration and Immigrations Enforcement scrape data from social media to find undocumented people or advocates for causes the admin dislikes. Polling organizations, journalists, activists, and claimed "illegal terrorists" are having their online speech and activity used to reliate against them.
When the internet and online social spaces are rife with surveillance, and our online activity is so often tied to our personal identifiers (IP address, name, phone #, email, address, etc.), the job of a malicious government is easy. Every time you fedpost about Trump on Instagram, express support for groups the US is opposed to, or write a controversial article with your name at the top of it, you leave a trail of breadcrumbs--nay, loafs--that lots of organizations and governments can use to learn your political persuasions, and potentially persecute you for them.
When you can't trust your network, your ISP, the sites you're visiting, or your government with your internet activity, the value of anonymity becomes clear. The Tor Browser is a web browser that mitigates many of these threats by making your browsing anonymous. Using it, you can create and maintain accounts on various websites, participate in conversation, publish your writing, and get access to information without being surveilled.
The first job of the Tor Browser to hide your traffic from your network, your ISP, and to keep your IP address hidden from the websites you visit, and it solves this problem in a very cool way. When you visit a website through the Tor Browser, your traffic is sent through the Tor Network, which is a network of thousands of servers run by volunteers. Your traffic is encrypted three-fold and sent through a randomly-picked three servers in the network, in such a way where no single node can see both your IP address the IP address you're visiting.
This means that your network and ISP can't know what website you're visiting; it'll just be a bunch of encrypted traffic into the Tor Network. The website you're visiting also won't know where your connection came from, just the IP address of the exit node. And even the tor nodes your traffic was routed through can't simultaneously know your IP address and the address of the website you were visiting. Your three-node "circuit" will automatically rotate periodically as well. Basically it's a proxy/VPN on steroids.
Now, as mentioned above, it's important to stress that though your traffic is truly hidden, by default your network and ISP *will* know that you're using Tor for /something/. Using Tor is not illegal anywhere in the US (as far as I know?), but it may be suspicious to your network administrator or possibly blocked on your network. If you're worried about your network and ISP knowing you're using Tor, you can use "bridges", which are a fourth optional random proxy your traffic will go through before entering the Tor Network to further obscure the nature of your connection. You can easily enable this in your "connection settings", which will be available right after starting the browser for the first time.
Also, because your traffic is being passed through so many servers on the way to its destination, keep in mind that browsing will be significantly slower than you're probably used to. It's comfortable enough on a fast home connection, but trying to stream HD video, or using Tor over a mobile connection or with slower home internet can get painful.
Aside from the network, online surveillance is often done through "fingerprinting", where a website can see/query your browser for all sorts of information to build a profile on your connection. The fonts installed on your computer, browser cookies, browser extensions, your screen size and many more variables can be used to build a unique fingerprint and expose you to tracking.
To stop these kinds of attacks, the Tor Browser has many anti-fingerprinting protections build-in, that attempt to make your connection look like every other Tor users, so no-one seems unique.
The Browser uses letterboxing, which is a curious little feature that disguises your screen size by having a certain number of pre-chosen website sizes that the window will snap to. This is hard to describe in words but you'll notice it quickly when you use the Browser. You can resize the window as granularly as you wish, but the website will only grow and shrink in certain particular sizes.
Every time you close the Tor Browser, all cookies and history is removed, so you'll get a clean start every launch.
When information about your browser and operating system are typically sent to website, the Tor Browser will lie and claim every user is using the same devices. It will hide your time zone, your installed fonts, and refuse to use many risky APIs that can be privacy-intrusive.
By default, the Tor Browser will use these network and anti-fingerprinting features to make your browser and your connection look as similar as possible to every Tor user, so everyone's traffic is all mingled and indecipherable and difficult to track, but you can definitely break your anonymity by making mistakes when using it. Here are some things to avoid:
This is the most common class of mistake Tor users make that leads to arrests. Always understand what information you may be accidentally linking together that could connect your anonymous activities to your personal identity.
Because the Tor Browser is designed to make everyone's connection look similar, if you start changing settings or installing extensions, your browser will become more unique and track-able. Almost always try to just stick to the defaults!
There are a couple notable exceptions to this rule; options that are meant to be played with if you so desire:
In the browser settings, there are three "security levels" you can choose from. Choosing the "safer" options will restrict websites from more potentially-risky activity, at the cost of many more websites not being able to function. I'd recommend defaulting to the most secure option and lowering it if a particular site demands it.
As mentioned earlier, you can optionally use a bridge to hide the fact that you're using Tor from your network and ISP.