These terms about the Internet get misinterpreted sometimes.  

Basically, the entirety that is the Internet is made up of individual websites (like this one) indexed on servers across the globe.  I have a WordPress site, others have Wix or Godaddy.  So the servers may be different but we all end up on the same web.  

The term Deep Web is that entirety.  Live Science has estimated that over 8 billion pages take up around a few thousand petabytes of data on the deep web.

Here’s the kicker: only 3-4% of the Internet is Public.  About 97% is password protected.  This includes your company VPN, cloud services, and the back portal of my WordPress site- to name a fraction. 

The Dark Web is a section of the Internet accessed by sites like Tor.  Lots criminal activity, hackers, sex trafficking (horribly so) and drug running (read up on Silk Road).
Most of these sites can only be accessed by invitation.   You have to know someone.  But you wouldn’t want to anyway.  

You could illegally download movies, but would have to sift through a bunch of nightmarish content I can’t even say without choking up. 

Take it from me.  Stick with Google and the 4% of the indexed web.