In a standard stereo setup, people typically use receivers, but at the higher end the various functions of a receiver are spread out amongst individual components. In a high-end stereo setup you might have a separate tuner for radio, a separate CD player, a Separate pre-amp, a separate speaker amp, etc. In that sense a pre-amp is generally the device that handles volume control, switching between the sources, and often times a headphone output. In a regular setup an “amp” is simply the power amplifier that powers the speakers.
Then you have headphone amps, which unlike standard stereo components which are tailored for use with speakers, are tailored for use with headphones. Typically stand-alone headphone amplifiers offer better headphone sound quality than using a preamp or some other component where headphone output was merely an afterthought.
Where things get complicated is that some headphone amps offer the ability to also act as a traditional pre-amp in a home stereo. In this case you could simply hook up a traditional speaker amplifier.
Here you can see my Little Dot II+ which is a headphone amp that
also has the ability to act as a preamp. When I use it as a preamp, the pre-amp outputs on the LDII+ are connected to the inputs on my stereo speaker amp which can be seen below. The black Yamaha is an example of a traditional “amp”.