Tor

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Tor (The Onion Router) is "a network of virtual tunnels that allows people and groups to improve their privacy and security on the Internet. It also enables software developers to create new communication tools with built-in privacy features. Tor provides the foundation for a range of applications that allow organizations and individuals to share information over public networks without compromising their privacy"[1].

Hidden Services

Tor not only lets an end user connect to servers on the "clear internet" via exit nodes, but also lets nodes (including desktop computers) proxy services (TCP ports they're listening to) as hidden services (also known as onion services), where the entire route between user and service is hidden from tracking and encrypted end-to-end.

Silo Examples

Facebook is available at https://facebookcorewwwi.onion – quite remarkably with a trusted HTTPS certificate for the .onion address! An article on the Tor blog describes it in detail.
Keybase is available at http://fncuwbiisyh6ak3i.onion

IndieWeb Examples

unrelentingtech's https://unrelenting.technology is available at http://fusrodahw6jpucw6.onion
Marty McGuire's https://martymcgui.re/ is available at http://schmartyp7qtjzn7.onion/
AJ Jordan's https://strugee.net/ is available at http://strugeefndmsxrgmb4zbebvznkaz42khllpjuvbzuhzkrnwwt3lt5pad.onion/

v3 `.onion` Addresses

As of TorBrowser 8 v3 addresses are supported. They are 56 characters long:

ebkmsk4buv7scyggulmw66gl5u5337zawjwomiiyzaufo3mu6jhrqxid.onion

Mining

You can use software to search endlessly for a particular desirable prefix or prefixes. mkp224o is one such program.

Given a 1-CPU Digital Ocean droplet 60 days of mining for a very limited set of terms (15 total; couple terms each at 6, 7, 8, 9 digit length):

6 digits, several finds per day
7, several per week
8, one in last 60 days

The more CPUs you throw at the problem the more likely you are to find what you are looking for.

Canopy is using mining and Canopy#short-hashes to render v3 addresses as usable identifiers.

Tor Browser Bundle

For end users (as opposed to servers running relays), it is recommended to use the Tor Browser bundle since it comes with a pre-configured web browser to protect your anonymity, and is self-contained[2].

While Tor Browser is running, it also servers as a local node and enables users to launch ad-hoc hidden services (e.g. OnionShare) or persistent ones [where the user retains the private key to a .onion domain that can even be publicized] (e.g. HiddenID).