|You need to change the display resolution in Windows to match the new monitor's native resolution (ie 1920 x 1080).|
Control Panel >> Display >> Adjust resolution
If there is no 1920 x 1080 in the list of available modes, you'll have to either change the graphics card or put up with the black border.
Another option is to "stretch" the existing image to fill the screen, using the monitors own vertical and horizontal width adjustments which is usually through an on-screen menu-system using the monitor's buttons. However, this can cause distortion of some elements in the displayed image.
That's why it's better to set the resolution to the correct values first, then adjust width & height only if necessary (which is very rare if the resolution is correct to start with).