blockchain fork – What exactly is forking in cryptocurrency?

My understanding was that forking is essentially update made to the blockchain software associated with the cryptocurrency, until I came across this:

A “fork” is the term used to describe a single blockchain diverging into two paths. Generally this occurs as the result of a significant change in the network’s protocol that effectively splits the blockchain into an old way of doing things and a new way of doing things.

What exactly does “diverging into two paths” mean? Should the whole blockchain be updated after the forking (no matter whether the forking is soft or hard)? (Which seems to be the case according to the description in the same section on the webpage.)