According to Merriam-Webster on-line -
Etymology: by shortening
Amp, in the popular form is the shortened form of: Amplifier; specifically: an electronic device (as in a stereo system) for amplifying voltage, current, or power
Why this term matters: An amplifier is a device that increases the size of a signal. The amp is the last stop for the audio signal before it goes to the speakers. More power, usually expressed in watts, means louder sound because the quality and size of the sound wave signal size is better. Beware of the tricks manufacturers can play to make their amps seem better than they really are [although the article is about car stereos, the same principles hold true—and it's a good article for explaining the basics].
Receivers are a combination of a control (pre-) amp, amplifier and tuner all in one. Integrated amps are receivers without the tuners. A third type, separates, require an additional component to do the switching and volume adjustment that receivers and integrated amps do as part of their set up. Separates are considered the ultimate in amplification.
Keep in mind that all three have cooling needs, which you should consider when setting up your system. It is up to your budget and your ears to decide what amp is best for you.