Diaspora* is still in alpha, so explanation by @ryan_blockey is helpful.
To begin, let’s look briefly at how Diaspora works. The platform is focused on giving users ownership of their data. Your data, including your profile, your posts and pictures, extensions and applications you’ve connected to D*, are all your seed. Your seed is located in a pod, which is just a server. So far this is all just like Facebook or Google+. The difference is that there are many pods within Diaspora. You can host your own, in fact. Or you can just create a seed on a friend’s server-pod. Or there are public pods you can join.
It’s actually quite similar to the way WordPress works. My blog is hosted on WordPress.com. Other people use the WordPress platform, but they host their blog on their own servers or they pay for hosting on someone else’s servers. Your Diaspora seed is your personal social space on the web and you can keep that information wherever you want.
So that is how Diaspora is built and implemented, but how does it work in the larger scheme of the web? Diaspora is eagerly integrating other social networks. You can already post to Facebook and Twitter from your D* page. Your contacts on D* are organized into Aspects, which are more functional versions of G+’s Circles. This extra functionality allows you to make Aspects public and you can even follow someone else’s Aspects. Or you can keep them private. D* has also implemented hashtags, à la Twitter. But users can follow hashtags just like people and be part of a conversation across all of Diaspora.
One of the major advantages of Diaspora is that users control and own their data.