how do I operate two applications in two different monitors at the same time?

You need two video cards. Windows XP and Windows Vista automatically support this as long as you have two video cards connected to your motherboard.

If you're using another operating system you're going to have to update your question because I can't make it general enough to account for all OSes.

Make sure you have available slots (PCI, PCI-X, or AGP) before you do the upgrade. It shouldn't be a problem how you mix and match the cards or whether or not the video card is built in.

This only applies to desktop systems and not notebooks.

There are special VGA splitters or USB devices that also allow you to use multiple monitors but they are often very expensive and low quality.

Your best bet is two video cards.

Once you have the hardware installed, you'll need to go to DISPLAY settings and activate the other video card and monitor. Simply check "Extend my Windows desktop onto this monitor" and you're done. Set the resolution and position as needed.

On the newer notebooks, you can use the VGA port either to mirror your notebook screen, in lieu of, or as a dual display. To activate the dual display (assuming your notebook doesn't prompt you when it detects a monitor or you don't have some utility) it is the same as the above with a desktop with two video cards.

Just wanted to mention that most modern video cards support dual displays, no need for to video cards.