Say you want to send a letter to a friend of a friend, but you don't know their address so you can't send it direct.
Instead, you can package your letter inside a second letter to your mutual friend, asking them to finish filling out the mailing label for you so your message will reach the intended recipient.
They call it "Onion Routing" because the message can be wrapped in multiple layers of these routing requests, with each recipient stripping off the layer addressed to them and forwarding the remainder on to the next connection in the chain.
Using this protocol, so-called "Friend to Friend" networks can still enable communication between non-friends so long as a "5 degrees of Kevin Bacon"-style connection exists between you and whoever you're trying to reach.