The Dark Web is just a set of different networks that work with peer to peer connection, and most of them do work with encripted data. They make use of VPN connections for hiding the real peer's IP address from each other, hence making it a safe and anonymous connection.
These networks' contents are not indexed by search engines, and the famous ones are usually open source because of the safety brought by open code.
Onion is just one of these many networks. It is routed by TOR (The Onion Route), and it just works as Toni mentioned but not through servers, through other peers, actually.
There are a lot of routers and peer to peer networks out there. The most known are Freenet, I2P, Zeronet, Stealthnet, Global leaks, Onion (the most famous and mentioned earlier), and many many others.
Yes, it is possible to access the dark web in a way other than with TOR.
Is it possible to access content from The Onion network without TOR (It is open source so others may use its code)? No.
It's the connection between the sites.
www has a basic TCP connection from your IP address to the server IP address.
Dark web routes your connection through multiple servers around the world. Dark web is all about the tor browser which makes this connection for you.