A display screen made with TFT (thin-film transistor) technology is a liquid crystal display (LCD), common in notebook and laptop computers, that has a transistor for each pixel (that is, for each of the tiny elements that control the illumination of your display). Having a transistor at each pixel means that the current that triggers pixel illumination can be smaller and therefore can be switched on and off more quickly.
TFT is also known as active matrix display technology (and contrasts with "passive matrix" which does not have a transistor at each pixel). A TFT or active matrix display is more responsive to change. For example, when you move your mouse across the screen, a TFT display is fast enough to reflect the movement of the mouse cursor. (With a passive matrix display, the cursor temporarily disappears until the display can "catch up.")
A more recent development is organic thin-film transistor technology, which makes it possible to have flexible display surfaces.