What is Hubzilla?
This is already explained in Hubzilla's help section, but maybe that's too technical, so I'll try to simplify things a little and also explain it in relation to more traditional social networks.
Facebook, Twitter, the recently closed Google+, and similar services have one thing in common: They are all run by a single entity. One company has control over it, and, in extension, your data. They don’t connect with each other, and if one of those companies goes belly up or simply decides to shut down a particular service, as in the case of Google+, the users are out of luck. Also, if they are offering the service “free of charge”, they have to find other ways to generate income, and that is usually by monetizing your data. At best, you need to trust that the corporation behind the service doesn’t abuse its access to all that personal information. At worst they might become the target of sophisticated hacking resulting in a data breach. The larger the user base, the higher the attraction to criminals.
Hubzilla is different in that there is no central server hosting every user's data. It's made up of a network of individual servers which all communicate with each other. You can look at it in a way similar to e-mail: You can send an e-mail from firstname.lastname@example.org to email@example.com, despite the fact that Google (GMail) and Yahoo are two different companies, each with their own servers. That's because e-mail is based on open standards and protocols, which allow separate computers, or servers, to talk to each other.
Everybody (with the technical skills and necessary resources) can set up their own instance, in any desired size: from a small, single-user hub on a RaspberryPi or MiniPC, to an instance on a dedicated server suitable for hundreds of users. The size of the userbase of any individual server is unimportant, since they all communicate with each other. In order to connect with people on other servers, all you need to know is their "webbie", which looks similar to an e-mail address. This channel's webbie, for example, is "firstname.lastname@example.org".
This decentralized approach has several advantages over the proprietary services. For one thing, you have a choice of where you create your account, and who to trust with your data. You could even host your own. It also makes the network more resilient to outage, especially with Hubzilla's Nomadic Identity feature, on which I will elaborate later. It also means you're not giving up all your private information to advertisers, which is the business model of the proprietary networks.
In addition to the social network capabilities, Hubzilla also offers tools like cloud storage, calendar and address book functionality, a blog (the articles app), a wiki, and web pages.
Hubzilla is part of a larger world of decentralized networks that includes other projects which, to some extent, can also communicate with one another. Together these individual projects make up the Fediverse / the Federation. I will introduce some of those other projects in a dedicated article.