The die is an itty bitty cube of semiconductor, the composition of which determines the color of the light given off.
It sits in the bottom of the die cup, which has reflective sides to reflect the light emitted by the die toward the dome end of the LED. T
he epoxy body is shaped to act as an inclusion lens and focus the light into a beam. The distance from the die cup to the domed end of the lens determines how tightly focused is the resulting beam of light.
Some LEDs have flat or even concave ends to dispurse the light into a wide beam.