If you've ever used email, you've used a federated service. Basically what federation means is that the network, instead of being composed of servers controlled by one company like Twitter or Discord, is composed of many independent servers communicating with each other over an open protocol, allowing anyone with an account on one server to communicate with people on any other. For example, a gmail.com user can talk to people on proton.me, or a Mastodon user on tech.lgbt can follow a Sharkey user on transfem.social.
Generally with federated networks, instead of plain usernames people get two-part addresses, composed of their username and the domain of the server they're using. For example, johnsmith@gmail.com has an account on gmail.com with the username "johnsmith", or @root@isopod.zone has an account on isopod.zone with the username "root". When you send a message on a federated platform, you're submitting it to your server, which then relays it to the server of the recipient(s), which then relays it to them.