Edit: I appreciate everyone who took their time to comment and provide explanation. All the comments were really helpful and informative. Thanks.
The main idea of the fediverse is that there is one standard (eg. lemmy), with many servers running a copy of that standard with their own settings (instances). These instances all interact in a way that it seems to the user like a single server (federation). At times the owners of each instance will want to block other instances (defederation).
federation is a trust relationship.
In the context of fediverse, is server operators allowing other server operators to read and write between each other.
For example If Nextdoor federated with MySpace, a Nextdoor account could post something and a person on MySpace could see it in their feed.
The explanation given here are very good. I just want to had add that theoretically, any Lemmy instance can communicate with another instance from another fediverse software but in practice, developers haven’t had the chance yet to make sure every feature is compatible with the each every other software.
If you want to try sending message across the different instances and softwares of the fediverse without spamming or flooding people discussion spaces there is a dedicated community: !testfediverse@jlai.luSo to run lemmy you create your own site, and then every other site that runs activitypub (the underlying engine) you can choose to download and mirror the data from them, with any changes automatically being sent to your instance. This is federation. If one instance starts posting stuff that is immoral, illegal, or just generally goes against what you want to have on your instance, you can choose not to download stuff from them any more. This is defederation.
Because every site running activitypub uses more-or-less the same underlying language, it means you can federate with instances that aren’t the same and they’ll more-or-less function. This is how mastodon threads are capable of sometimes popping up in lemmy posts.
