Disclaimer: This article is for informational purposes only, it may neither be used for illegal activities nor will I be liable for any kind of data or privacy compromise and/or monetary loss that you incur in doing so.

For most of us, internet starts with E-mail providers; in-between we browse some websites like Facebook, YouTube, Reddit & Quora and ends with Google but the internet doesn't end there. Almost every organisation or company has their own website and their databases are accessible online.

Deep Web

Deep web and Dark web are often confused and used interchangeably. Although it seems pretty obvious we, humans seek affirmation.
Deep web is anything that cannot be indexed by search engines for limitations of crawlers, password based access restrictions and whatnot.
Q: How can I access the deep web?
A: You already do. Well, not the entire deep web just the part you have access to. You login to your email account, those emails you see there are a part of the deep web so are your Facebook messages, Twitter, Instagram, and even rocket chat's DM.
Basically anything that requires authentication to access the data is stored in deep web.
The deep web is HUMONGOUS. No wonder why iceberg analogy is applied to it.

Steemit also comes under surface web besides Google. Black: Surface web; Blue: Deep Web; Red: Dark web

Dark web

Dark web is purposely hidden from the search engine crawlers and unlike regular websites their domain ends with .onion suffix.
When we hear a news about database of a website being compromised, it simply means that the hacker was able to gain access to the database(deep web) by exploiting the security vulnerabilities and extract the data.
After that, he'd use the data to his own benefit and if that data is irrelevant to him he'd sell the data to relevant buyer(s) and make a profit but setting up a website or blog on the regular web for that purpose is risky and can get both seller and buyer busted.

Apparently, I use this website for hosting images that I use here.

This is where the dark web comes into play. It allows users to browse the Internet anonymously and securely. Websites set up there are also anonymous unless the owner decides to reveal their identity.
How can I access the Dark web?
A: Download Tor bundle and you're good to go!
[Note: Read the instructions found in the download page carefully before using tor and do not maximize it's window.]

Onion websites(I didn't hyper-link them so you'll have to copy into a text file and browse later) [Disclaimer applies] [Added security: use a spare computer or virtualbox, put a tape on your microphone and webcam, create a separate user-account with restricted access; don't use or original name, email username on the darkweb, even better, don't signup anywhere]:

The Hidden Wiki 1: http://zqktlwi4fecvo6ri.onion/wiki/index.php/Main_Page
The Hidden Wiki 2: http://hwikis25cffertqe.onion/index.php?title=Main_Page
Tor Links(moderated): http://torlinkbgs6aabns.onion/
Torch(Tor search engine): http://xmh57jrzrnw6insl.onion/
DuckDuckGo(another search engine): http://3g2upl4pq6kufc4m.onion/
Ahmia(yet another search engine): http://msydqstlz2kzerdg.onion/
Imperial Library: http://xfmro77i3lixucja.onion/
Liberated books and papers: http://52wdeibt3ivmcapq.onion/
The secret story archive: http://tssa3saypkimmkcy.onion/
Lobby: Truth Seekers: http://zdobrh7ncrwttnjq.onion/
Russian Wiki: http://doe6ypf2fcyznaq5.onion/
Deep Tune: http://tune4xs6mj2evcr6.onion/
One of the biggest tor directories: http://bdpuqvsqmphctrcs.onion/noscript.html
This is my personal Favorite: http://retardo6vlfxyzen.onion/

Sources:

Quora
PC Advisor
.onion links: from my own research & http://52wdeibt3ivmcapq.onion/

Further reading:

My visits to the darkest sites on the deepweb: reddit
Deepweb story that happened to my friend: reddit
Hilarious yet informative: 9gag

Safe journey.. err.. browsing!

Regards,
@onion