Connecting 21" screen to LVDS connector on motherboard
You need to disassemble the monitor to get the LCD panel out. You won't need the power supply board or the HDMI/VGA/whatever controller board. (carefull of high voltages, even unplug from power, the capacitors of the power board can kill you.)
The monitor panel has (probably) a LVDS interface.
Your laptop connects to LCD using a LVDS interface (if it's an old one from before ~2012)
or a eDP if it's a newer laptop. If it's eDP you will need LVDS to eDP converter board, which are very uncommon.
If it's LVDS you just need to do cable work to connect the interface correctly.