Displays come in many sizes and resolutions, with a few different technologies. AMOLED is a common display type providing fast display updates with low power. This type of display is essentially a matrix of pixels that can be switched on or off.
The display has a digitizer in front of it. A digitizer detects the position of a finger on the display and sends this information to the operating system (which provides this information to an app). A common touchscreen technology is capacitive. When a finger touches this type of display it conducts a tiny amount of electricity, resulting in a distortion to the digitizers electrostatic field which can be measured to give a position on the display.
The digitizer can be overlaid on the display or integrated into the display. For example, “Super AMOLED” is a common display with an integrated digitizer. Digitizers support a certain number of simultaneous touches, 2 and 5 are common.
To help protect the display glass from scratching, the glass can be toughened during manufacture. Gorilla glass by Corning is commonly used in phones (there are currently 5 versions of Gorilla glass, 1 – 5 (for example, later versions improve scratch resistance).