Reblogging has been around for a couple of years now on WP.com and users are heavily divided about it. The biggest problem is certain "bloggers" who create entire blogs of only reblogged content. I report them as spammers whenever I find them. If someone reblogs a post it also transfers all media files you've added to that post to their media library.
But you can disable it to some extent. If you go to Settings ->Sharing in the dashboard you'll find the option toward the bottom to hide the reblog button in the admin bar. However, this will not prevent people from reblogging your posts directly from the Reader, so you can't disable it completely.
As has been mentioned, it only publishes an extract with a link to your site, and you can choose whether you want to approve or delete the pingback that's created to the reblogger's site. I also have a notice on my site asking people not to reblog, for what it's worth, but the Terms of Service actually state that you give permission for your posts to be reblogged by using the site, so there's little we can really do if someone reblogs something and you definitely don't have grounds for an official takedown request.