I strongly suggest following ForumAccount's advice unless you have a specific reason for sticking with VB.NET 2003, e.g. that's what your work requires or you need a feature that VB Express doesn't offer. Otherwise you're just limiting yourself unnecessarily.

As suggested, the functionality you seek was built into the NotifyIcon class from .NET 2.0. Prior to that there was a third-party solution available created by a developer named Hans Blomme. Since so very few people still use .NET 1.1 it has become hard to find. I searched and could find references to it but couldn't find anywhere to download it. If you must use .NET 1.1 then it might be worth your while scouring the web for it.