An optical amplifier works just like a laser, but with a few small differences. In a laser, light is created, usually due to spontaneous emission from the gain medium. Light then circulates in an optical cavity while being further amplified in the gain medium. A single photon can traverse the laser cavity many millions of times before being released.
In contrast, an amplifier does not (ideally) produce any laser output on it's own. An amplifier consists of a gain medium, just like a laser, but has no optical cavity in which light circulates. Instead, light enters one end of the cavity, traverses the gain medium only a few times (in the case of a fiber amplifier, only once) and then immediately exits.
In the case of telecommunication equipment, the input light is the data signal that needs to be amplified by the relay station. In many other applications, coupling a low power seed laser with an optical amplifier can be a convenient and efficient way to get a high power laser beam. A lot of laser cutting and machining is done this way, for example. In this case engineers may call the laser system a Master Oscillator/Power Amplifier or MOPA system.