What you will need:
a soft, anti-static cloth (for the plastic casing) (if required).
microfibre* (lint-free) cloth (similar to eyeglass cloths).
suitable cleaning fluid for the monitor's plastic case.
suitable LCD screen cleaning fluid or distilled water**.
* microfibre cloths can be found in automotive and electronic stores. A good substitute would be the soft cloth used to clean eyeglasses, or in an emergency a very clean oldpure cotton t-shirt. Never use paper products like tissue!

** for some screens the water may be mixed with isopropyl alchohol for stubborn marks, BUT always check the manufacturers instructions for your particular model. If in doubt and suitable LCD screen cleaner is not obtainable, use just water (preferably distilled).
If you are going to clean the monitor casing, do it before you clean the screen. Spray the anti-static cloth with a suitable cleaner (don't spray the fluid directly onto the monitor/laptop) and wipe the monitor's casing (be careful not to touch the screen with the cloth or cleaning solution).
To clean the screen, gently wipe the dry microfibre cloth across the surface of the screen (in straight lines) to remove any dust.
If there are stubborn marks, put some of the LCD cleaning fluid (or water) onto themicrofibre cloth and gently clean the area (never spray fluid directly onto the LCD screen, always spray/apply onto the cloth)
Ensure the screen/monitor is completely dry before use.

Cleaning a CRT (Cathode Ray Tube) monitor

What you will need:

a soft, anti-static cloth.
a lint-free cloth.
screen cleaning fluid (glass cleaner is usually ok).
CRT (Cathode Ray Tube) monitors can be cleaned in much the same way as conventional tube televisions.
Use the anti-static cloth and a suitable cleaner for the plastic casing of the monitor.
Spray the screen cleaning fluid onto the lint-free cloth (never spray directly onto the screen) and wipe the monitor's screen in straight lines.
Ensure the monitor is completely dry before you reconnect the power