probably should have copied the forum bbcode... regardless- the third system tray icon from the right is your network status, which appears to be wired. Why are you trying to diagnose the laptops wireless issue with your desktop? (If you are running SP1 or Non-SP XP I apologize since the icon will look the same)
On another note, you have WAY WAY WAY too many system tray notification icons... this means you have a lot of background programs running. I, for example, have four at the moment. MSN messenger, the "safely remove" icon, volume control, and taskSwitchXP Pro.
If you are willing we could help you remove the cruft, but for now- as I said before, this appears to be a screenshot of your desktop PC, not the laptop encountering issues (correct me if I'm wrong).
Im not using wrieless on this computer
well, that would explain why it's wireless connection doesn't work.
the screenshot we want would be the computer experiencing problems. Since you refuse to reboot the router and almost certainly put a end to this whole mess, we can go no further in resolving a router issue. As such, we can only assume (read:hope) that the laptop's wireless perhaps has a problem.