These aren't official, just general internet slang:
The stuff you browse everyday: YouTube, DuckDuckGo, Facebook, etc.
Level 1 - Surface Web
This level is still accessible through normal means, but contains "darker" websites, such as Reddit.
Level 2 - Bergie Web
This level is the last one normally accessible: all levels that follow this one have to be accessed with a proxy, Tor or by modifying your hardware. In this level you can find some “underground” but still indexed websites, such as 4chan.
Level 3 - Deep Web
The first part of this level has to be accessed with a proxy. It contains Hacking, Adult Content, etc. This is the beginning of the Deep Web. The second part of this level is only accessible through Tor, and contains similar stuff.
Level 4 - Charter Web
This level is also divided in two parts. The first can be accessed through Tor. Things such as drug and human trafficking, banned movies and books and black markets exist there.
The second part can be accessed through a hardware modification: a “Closed Shell System”. This is serious. This part of the Charter Web contains Illegal Content, experimental hardware information ("Gadolinium Gallium Garnet Quantum Electronic Processors"), but also darker information, such as the "Law of 13", World War 2 experiments, and even the location of Atlantis.
Level 5 - Mariana's Web
A place where AIs come together and comment on cat videos. You need access to quantum computers to see this stuff...