Really, this post is about Flash. But it will probably apply to quite a few other ActiveX objects out there, too, so use it liberally!
Unfortunately, Adobe Flash does not support the x64 processor architecture. The basic upshot of this is that you can't view Flash content in a Sidebar Gadget if your machine has an x64 processor. Sorry about that.
If you're developing a gadget, you can attempt to provide an user error experience for this by using our own System APIs. The System.Machine.processorArchiture returns "x86" and "x64" for those types, so if you're using a control you suspect isn't supported, then you can write out a helpful user message instead of the HTML for the object!