What is the definition and purpose of a pre-amplifier?

There are many reasons why you would want a pre-amp and a power amp. The easiest to understand is when your source and destination are far away from each other. In this situation a pre-amp can be helpful so that the noise that is picked up on the line to the power amp is minimal compared to the signal itself.

Another situation is if you were going to be performing some filtering on a signal. All of the filtering elements can add noise to your system and by adding a preamp the noise in the filtering is minimized compared to the signal. Also the preamp can act as a simple buffer between your source and the filtering equipment.

And yet another reason you would want it is because your preamp isn't able to provide the current gain required for the power amp. You may still wonder why not just use a power amp? Power amps are more difficult to directly change the volume on while a preamp is much easier. So you can change your volume on the preamp and have a fixed gain on the power amp.

Simplistic view: There are N different kinds of pre-amplifiers and M different kinds of power amplifiers. For an application, you can pick one of the N pre-amps and one of the M power amps, instead of having to find a combined unit that is one of M * N.

Think about pro audio for instance. There exist microphone pre-amps. A mixing board will have a whole bunch of them, one for each mic input. It would be silly to replicate that in a power amp. The power amp just gets the mixed result. In fact there may be more than one power amp. One for the "house", and a different one for local monitoring.

A microphone pre-amp is a different beast from, say, a guitar pre-amp, etc.

Why would a guitarist want a separate pre-amp and power amp? Well, not all do. There are "combo" amplifier that have a preamp, power amp and speaker cabinet all in one. (And there have been electric guitars with a built in speaker, yielding complete integration.) But having separate units in a rack: pre-amp, equalizer, effects processor, power amp, gives you flexibility.. Don't like how the power-amp drives the speakers? Swap it out; see how the setup sounds with a different one. You can change one "variable" at a time in search of tone.

A preamplifier processes a signal to make it fit for the next stage in the signal chain.

A stereo set 20 years ago would have a phono preamp. Other signal inputs than phono would have a sensitivity of for instance 500mV, while the record player only would give 3mV. So you needed a preamp to pump this up to 500mV. That same phono preamp would filter the signal because the phono's frequency characteristic wasn't flat. That was the infamous RIAA correction. So the preprocessing isn't always limited to changing levels.
A microphone preamp also has the function of amplifying the low input voltage to a more common level.

Further in an audio amplifier you'll have a preamp to create the required output voltage for the speakers. This will drive the output stage which supplies the required current/power. Amplifying voltage and current may be separated.