A computer screen can be glass or plastic depending on the design. In older monitors, the screen is a key part of the display. In newer monitors the actual display is inside the monitor and the screen itself is merely for protection.
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A CRT monitor works in a similar manner to an old-fashioned television. It has a glass screen coated in dots of red, green or blue phosophor. The monitor aims an electron beam across the inside of the screen, covering the whole screen, one row at a time. Depending on the image, the electron beam varies in strength, depending on which dot it is passing over. This happens fast enough that the user appears to see a solid image with some dots illuminated and others not, thus creating the picture.
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An LCD monitor also breaks the picture down into dots and recreates it. However, it works by shining light through two sheets (usually plastic) with a layer of liquid crystal between them. The sheets are polarized in a perpendicular manner, meaning that the light which passes through the inside sheet cannot pass through the outside sheet without manipulation. This manipulation comes from the individual crystals which, in response to an electrical charge, can change the direction of the light passing through and, in turn, cause it to pass through the outside sheet and become visible. The actual outside of the monitor can be glass or plastic and is simply a transparent protective layer to prevent damage to the polarized sheets. An LED screen is simply a version of an LCD screen that uses LEDs to shine the light through the polarized sheets and crystals.