The stimulated emission stimulates more emission, so there is a rapid, exponential growth of photons in the doped fiber. Since most systems still work at 1310 nm, considerable research has been done to find materials that would work in this range. The basic structure of an EDFA is very simple. Just like in a laser, the emitted photons then stimulate other emissions, so there is an exponential growth of photons. Supporting the amplifier is a pump laser, which supplies the amplifier's energy, a coupler, which combines the pump laser beams and the signal laser beam and puts them on a single fiber, and an optical filter, which removes the remaining traces of the pump beam so that it doesn't interfere with reception of the signal.
Erbium has several important properties that make it an excellent choice for an optical amplifier.
That gives them the ability to amplify signals in a band where high-quality amplifiers are most needed. Erbium's quantum levels also allow it to be excited by a signal at either 800nm or 980nm, both of which silica-based fiber can carry without great losses, but aren't in the middle of the signal wavelengths. Those bands are also far enough away from the signal bands that it is easy to keep the pump beam and the signal beam separated. This property is extremely important, because the quantum efficiency of the device is dependent on how long it can stay in that excited state. If it relaxes too quickly, more photons are needed to keep it excited, meaning more input power is needed to make the amplifier work.