In the olden days email worked in a quite similar way, although with different protocols. Before Gmail and things like that many individuals and organizations ran their own email servers. Fediverse addresses are also similar to email addresses: myusername@myinstancedomain.
Fediverse servers are usually called "instances". It's not really clear where this terminology comes from, but most likely it originates from object oriented programming.
The fediverse has existed for about as long as Twitter, starting at around the same time in 2008, but up until 2012 its existence it was centralized around a single instance called identi.ca , or in the form of StatusNet subdomains.
The word "fediverse" itself appears to have originated at around the same time as this split.
GNU Social was the main flagship software of the fediverse up until 2016. This period of time between 2013-16 might be called "The Quitter Era". In 2016 there appeared a fork called postActiv and a new system called Mastodon, written in Ruby. Then in 2017 a new fediverse server system called Pleroma appeared, written in Elixir.
Today Mastodon is by far the most popular fediverse software.
In terms of numbers of users there was probably a peak somewhere around 2010 on identi.ca. After that there were not many users (a few tens) until the first major Twitter exodus in 2016. Today the size of the fediverse has grown far beyond even the identi.ca peak, with hundreds of instances and many thousands of active users. By the end of 2017 Mastodon reported a million user accounts, although only a fraction of those will have been active.