Vlad
TOR TL;DR:
When the user wants to visit a website, they dont connect automatically to the website, but they encrypt and nest their data three times. Now the request is send over three different "nodes" run by volunteers through the network. Every time a node is passed, they decrypt one layer and forward it to the next node. Only the last node knows, where to send the data (and if using https not even the content). To the website, it is the last node who requests the data, not you with your personal IP.
[The onion thing comes from the layered encryption, like an onion.]
I think (but am no expert here, so please correct me!) that a so called onion service is not on the clear web. They have a different adress scheme (asdf.onion) and work this way:
When the third server in the chain sees, that it is an onion link, they don't connect to a normal website, but they route the data through the volunteer run servers to the website, which is itself "hosted ont he network"(?).