The battery of the VAIO VGN-FW590 allows the laptop to remain working when not plugged into a power point. Replacing the battery pack is not complicated and you should be able to replace it easily.
Turn the computer off, close the LCD screen, and insure the computer is unplugged from any outlet.
Turn the computer over so that the bottom of the computer is facing up and the "screen swivels" (DC in port and On button) are facing away from you.
Locate the battery pack between the two "screen swivels" at the top of the computer back.
Locate two sliding release switches below the battery pack.
Slide the "Lock" release to the right toward the center of the computer.
Slide the "Release" switch to the left, toward the center of the computer. It must be held in place because it is spring loaded.
While holding the "Release" switch toward the center of the computer, press on the raised portion of the battery pack away from you. The battery pack will pop up.
Once the battery pack has popped out of its compartment, lift it out of the computer.