I don't know what it is called, but it is either "LED" or "LCD".
You know the light on the desktop system unit.
I was wondering how to turn off the lights on the system unit when the computer is on.

LCD is for liquid crystal display (like your monitor) and LED is light emitting diode. I assume you mean the power lights on the computer case? You can't turn them off. If these are typical power on and drive activity lights, they each have a pair of wires that connect to the motherboard in what's called the front panel header. You can power down the computer, unplug from the wall, then open the case and pull the wires with no harm done. Just touch bare metal of the case first before reaching in. This puts you and the case at the same "potential" so there is no static discharge from your body when you reach in to pull the LED wires.

Now that the LED lights are answered.

Next up is LCD...

How big is a 22 inches monitor...

I need to know the length and width of it to see if I have enough space on the desk.

Here are inches to centimetres to make sure you are correct.

A few links here to browse also.

Have fun.

How big is a 22 inches monitor...

That depends on many factors - the most important being if this is a widescreen (16:9 or 16:10) or standard screen (4:3). It also depend on the stand. Some are fixed, some allow for 90° rotation, and some allow for height adjustments (which may be a factor if your desk has a hutch. Some stands are removable so you can mount the monitor on a wall. Also a factor is the bezel, the part of the frame that surrounds the actual LCD panel and holds it in place. Some are thin, some are very wide, often to accommodate speakers, or menu control buttons - or they may be wide because the maker feels that makes a fashion statement.

I just happen to use 22" Samsung WS monitors and these have a relatively thin bezel at 7/16". These monitors are 20 1/4" wide, and adjust from 16 1/4" to 19 1/4" in height (there is no "length" measurement with monitors).

A 22" 4:3 will be taller and not as wide.