How to use Two-factor authentication?

[AMSOS]

I've so far resisted the urge to go for two-factor authentication. I saw it work on a friend's laptop and it seemed there was not much control she had over how it works - a window just pops up with the code that you have to enter.

I now want to synchronise iMessages to my new phones and also feel I could set it up.

My question is simply this: How does it work when there is no internet connection. Say, I am logging into iCloud from a friend's computer for the first time. I don't have my own computer and my phone network is acting up. How will I get the code that let me login?

I've been using the Duo Mobile authenticator app with Gmail and my university email accounts and it works quite well. Does Apple use the Duo or a similar app?

 

[DeltaMac]

iCloud, by its nature, is an internet service (requiring an active internet connection)

More info: https://support.apple.com/en-us/HT204152

You would NOT need two-factor authentication, but if you log in to iCloud on another computer, you probably would need two-step authentication (not the same as two-factor authentication).
The Apple support article explains the difference, and goes through what you need to log in to your iCloud account, along with a lot of other iCloud information. If you ALREADY use two-factor, then you may not be able to use two-step authentication, which is for older devices.
Apple is not using some random third-party app for the two-factor authentication. It's an internal security upgrade for the system, and is a primary part of the AppleID/iCloud security. It's built in to the macOS system.