Wirless networking

The red line through the icon indicates that that network adapter is disabled in the device manager. (You can enable it by right clicking My computer, choose Manage>device manager. But the realtek is used for your ethernet cable connection. The 1394 connections are network adapters used for network connections to printers, etc. probably left over from your work setting, and can be disabled for the moment.
You don't mention what OS you're using, but I'll guess XP, and provide another link for vista just in case....
In any event, it's going to be necessary for you to reconfigure your network connection to the router/gateway you're using at home to connect to your ISP. Here's a general wireless networking troubleshooter, in the event that you've already covered the basics. And here and here are articles/links that will give you more advanced guidance. (Also here.)
But here and here are links to basics: to help you set up and maintain a network in xp, and here's an alternative for vista.
Another excellent resource to help you answer questions and troubleshoot the connection is this site, which has specifics that deal with many common ISPs.
In the end, you may find the easiest thing to do is start at the beginning by setting up a new home network and working through each step to establish a new connection.

I hope this information allows you to resolve this issue. If you need further assistance, please post back with a comment to this thread.
If I've managed to answer your question or solve a problem, please take just a moment to rate this post....thanks!