Ok understood, my system is capable of that, as hornblower has said the graphics card must support dual monitor, which when looking at cards isn't always as obvious as you might think. For example this is what cnet says in a review of your card.
"The good: Inexpensive; flexible driver software; certain retail packages let you use two monitors at once."
http://reviews.cnet.com/ATI_Radeon_X300_SE_128MB_HyperMemory/4505-8902_7-31406674.html
The bit that was confusing me was the idea of having monitors as totally independent of each other, they are not, you only have one computer, and therefore one mouse, so the mouse cursor must be able to pass from one screen onto the next. the additional monitor(s) should be setup as extensions of your primary monitor with the setting in display properties set to "Extend my windows desktop onto this monitor"
When you open an application it will likely open on your primary monitor, but as I think it was Mat said take it down from full-screen to half screen or whatever that is called and drag it to the monitor you want it on, when you maximise it, it will fill this monitor and only this monitor..
I run a Radeon 7000 and a Nvidia FX5500, total of 4 outputs, but only use 3
I still don't see how you have icons on monitor 2 unless you dragged some there and thus don't have them on monitor 1 or you are running some sort of software that recreates another set on the second monitor. This suggests to me that your card does not support multi monitor and is simply recreating you desktop on the second monitor. By default all your icons should be on your primary monitor and only your wallpaper should be on the other,
In any case check the setting under Display properties > settings tab and ensure the option to extend your desktop is checked.
PS if you do need to get another card, check with a computer retailer on prices b4 rushing to buy a quad card, dual monitor cards are not that uncommon these days but quad cards such as made by Matrox are still a niche market item and you may be able to get 2 dual pci cards cheaper than one quad.