With standard WordPress installation, you can manage a single blog with multiple users.
However, with WordPress MultiUser installation, you can create or let others create blogs.
For instance, WordPress.com is a very good example of site run on WordPress MultiUser Installation, so that people can sign up for a free blog and Administrator can edit them all.
For what you've asked, you simply need a social networking platform. So you can install WordPress MultiUser with BuddyPress and disable new blog registrations and allow only user registrations. That way, you'll have a robust social network on your site.
However, WPMU is not recommended for beginners and you'll need to work on some codes while installing and editing.
I've installed the plugin, with varied success.
Which plugin have you installed? Do you mean BuddyPress Plugin? Well, BuddyPress doesn't work with standard WordPress installation. It only works with WPMU.
Hope this helped :P