An amplifier is any device or circuit which (basically) applies a multiplication operation to an input signal. For instance, the signal produced from a microphone is weak and tiny. It alone cannot drive the massive loudspeakers you might see (and hear) at a concert. So an amplifier basically makes an exact copy of that signal but much larger in amplitude. This strong signal (with large amplitude and from a low impedance source) is able to drive the speakers.
An op-amp cannot power a large speaker, but it might be used to amplify the signal from the microphone. Op-amps are low-distortion, low-noise, and have high input impedance which are all things you'd want in a microphone pre-amp. An op-amp might be used to cleanly boost a signal so that a high power amplifier can deal with the signal effectively. The high power amplifier might be a transistor amplifier (see: common emitter or common source amplifier). Oh and by the way, vacuum tubes were used before all of this op-amp and transistor stuff.
The audio example is just one of many. Wireless transmission relies on amplifiers to create a high power radio frequency (RF) signals. You might use an op-amp (or more specifically, an 'instrumentation amp') to detect tiny electrical signals produced by the human body. (think: ECG/EEG)
Hope this helps.