Other people have explained it mostly, but piefed is the software your server (called an instance in the fediverse) is running. Confusingly, your instance is piefed.social, but that’s just an address, like any other address. It could theoretically be anything.

Piefed uses a protocol called Activitypub, which allows other software the implements Activitypub to understand each other. Often they can be interoperable. For example, I’m on an instance running Lemmy, which is different from piefed, but we can interact. They happen to be similar enough (Piefed being created from Lemmy) that it’s basically seemless. However, both of these also can interact with Mastadon too, but it’s less seemless. If you see someone starting a comment with an @[user] then they’re probably from Mastadon, which works like Twitter.

As for what to call this? Your home is called piefed.social. The larger thing I call Lemmy, but at this point it’s Lemmy and Piefed. Sometimes it has Mastadon mixed in. At large, all federated services are called The Fediverse, but that includes ones not interoperable with ours too. Just this set of interoperable software should probably get a name, but I don’t know of one.