In short, the dark web is a silly name given to content/servers on the internet that are not indexed by any search engine.
So say if there is a web site out there called
www.whatthefuck.com, you can't google it, you can do any search for it. You can only type in
www.whatthefuck.com into your address bar to start attempting to access the site.
There is also a massive amount of data out there not accessible from your browser either and needs specialised applications to access it. With a browser you only see HTTP data but there is also IRC, FTP etc
Generally though dark web = sites you cant google.
And as said, the dangers of all your medicare info being available means its easy for someone to buy an ID of someone else and do some forging/scamming/identity theft. Worst case i have seen is of a woman in WA who's house was legally sold by someone else. Not good at all.