An open proxy is a proxy that allows you to connect through it to another system, without first authenticating you.
A proxy does not do store-and-forward, as a mail-relay does; instead, your commands are immediately sent to the target-system.
A proxy does not add tracing information to material it sends; instead, it maintains a local log, and forwards commands un-modified.
As far as I am aware, the only way to find out who is behind an abused open proxy is to obtain a copy of the proxy's logs from the administrator of the abused system.
To the best of my knowledge, there are quite a numbert of people who still think it is a good idea to run an open proxy. There are many websites advertising lists of open proxies; many of the listed proxies have since been secured or shut-down.
You can check an IP address to see if it is running an open proxy-server at Masaaki Nabeshima's web-based proxy-checker.