Hubzilla. This is what we are.

Q: What do you (e.g. what does your project) do? What is your "product"?

We provide software for deploying and managing hubs.

Q: What is a "hub"?

A hub is a webserver or website which connects with other compatible webservers/websites in "interesting" ways, providing:
  1. Decentralised authentication (cross domain single sign-on)
  2. Decentralised communications (public and private)
  3. Decentralised (cross domain) privacy of web resources
  4. Nomadic identity (members of a hub can relocate temporarily or permanently to another hub)
  5. Mirrored directory services for finding content and members of other hubs

----- For those who want to know more -----

Q: What kinds of webserver applications are supported/provided?

Out of the box we support public and private personal and group communications, blogging, custom webpage creation, event calendars, photo albums, file archives and cloud storage, chat-rooms, RSS feed aggregation and republishing, and more. Any of these services can take advantage of crosss-domain authentication to provide cross-domain privacy for any published object. These applications may also be extended via plugins and/or entirely new applications created. For example, e-commerce applications and the ability to provided connected "store-fronts" as you might find in a shopping mall are currently under development.

Q: How do I access a private object hosted on another domain? (E.g. a private object which I am permitted to access.) Am I required to enter a special password or install a "consumer secret"?

Click the link to the object from any connected hub.
That's it.

Q: Who would use your software?

Anybody who wants to build a website and desires to connect it to other compatible websites to achieve benefits of scale (in terms of connected members and existing content), without requiring a huge up front investment of time and money to build a large website and/or develop a global community from scratch.  Also anybody who wishes to share something privately (any digital content) with somebody else but doesn't wish to maintain an account on their website for the other person.

Q: Could I provide a "standalone" hub or even a private "cluster" of connected hubs that isn't connected to the global network of connected hubs, for instance to use inside my company or school or even as a personal blog?

Yes.