I'm going to assume that you are not managing and not running OS X Server here. While it's possible to set up a proxy server with OS X Server (with some added configuration work), your question seems to be specific to a client web browser configuration.
With an OS X client request from a web browser such as Safari or Firefox, the HTTP 407 Proxy Authentication Required messages usually indicate something that's been implemented by the local site administration or local network management. This can be a way to cache web content and also as a way to track and/or to restrict access to specific web sites.
If you're on a school or a corporate network or similar, please check with the IT organization for assistance here; with the folks that manage the network.
If you're on your own network here (and not using OS X Server or some other web proxy), then this might be something weird with some local nanny filter installed in your gateway, or with some ISP web proxy outside your local network — more details on your network would be useful here.
Here are some technical details of the HTTP 401 and 407 authentication mechanisms.