What does "1 Gang", "2 Gang", and so forth mean when talking about electrical boxes?

The n-Gang terminology is also used in the UK, but with somewhat different meaning.

• For sockets it is similar, except UK plugs are bigger and not normally stacked two on top of each other. So a 2-Gang is wider than a 1-gang and has 2 sockets. The dimensions are of course different from the US ones.
• Blank face plates are similar. A 2-Gang plate is the same size as a 2-Gang socket.
• For switches it refers to the number of switches. 1-Gang, 2-Gang and 3-Gang switches may be the same size, but have 1, 2 or 3 switch buttons.

In the USA, "gang" generally refers to the physical size of the circuit box. With switches and receptacles generally conforming to a standard width, the number of "gangs" indicates how many of these standard switches or receptacles the box can accommodate. This question is more thoroughly answered by Steve Jackson above.

As Tor Klingberg hints at above, however, on the international stage "gang" tends to refer to the number of circuits within a single box, irrespective of the size of the box.

Here are two UK-based websites (with pictures) that explain what a "gang" is:

https://www.diydoctor.org.uk/projects/what-is-a-gang-switch.htm

‘Gang’ describes the number of switches on the plate.

http://www.lightwiring.co.uk/lighting-circuit-components/light-switches/gangs-and-their-ways/

If your light switch has one ‘switch button’ on it then it’s a single (or 1) gang switch. If it has two ‘switch buttons’ on it then it’s a double (or 2) gang switch. etc. etc.

Here is a random Filipino website which has what we would call a "3-rocker, single-gang" switch in the USA. But in the link it's called a "3-gang" switch:

https://www.cherryhome.com.ph/products/9b0a5ce68b768e35bdecb7c3a57d47ad