One of the first things that we have to keep in mind is that an amplifier does more important things than just amplify the sound you produce on the guitar.
This is to clarify, as the title suggests, that we are talking about guitar amplifiers, and not just any kind of amplifier (those used in any PA system, for instance).
We have to understand that guitar amplifiers actually help you to produce the kind of music you want to produce, and do not simply amplify the sound to reach the listeners. Such amplifiers are an essential ingredient in the total musical system that you appear with on stage or use during rehearsals.
It is true that guitar amplifiers do amplify the output signals, but not only on a linear level. Linear amplifications are done by, for example, PA system amplifiers. In linear magnification of sound, the input signals are identically reproduced as output signals. In the case of guitar amplifiers, there is amplitude distortion also, that is to say, the sounds created on the guitar are willfully changed according to the kind of music being played. Such changes, when we hear them, have already become an integral part of the music and we perceive them wholly.
Guitar amplifiers are usually set to produce sound at a high gain, which in turn, affect the sound created by the guitar through feedback. From among the many examples, let us cite two: the hugely magnified sound from the speakers influences string harmonics; this sound also influences the natural decay of the vibrating strings by modifying its sound.
But what about the type of amplifier?
Amplifiers can be divided into two broad types: the valve type and the solid state.
Valve type amplifiers are the older ones and many still prefer these to the newer solid state ones.
Both have their advantages and disadvantages.
To begin with one prime consideration, that is, cost, the valve type is more expensive than the solid state amplifier.
The former is also costlier to maintain. You might need to change the valves annually in order to keep your amplifier in an A-1 condition so that it can be driven at its peak during professional performances.
We were earlier speaking about amplitude distortion. The word distortion obviously suggests that the sound produced would be distorted. This also implies that there might be more noise and less music. This is true. But then, the idea was to tone down the noises and make them sound musical or, in other words, make them blend with the particular style of music. (However, there were/are styles of music where the noises are produced deliberately and they are left to remain noises.) Many guitarists also crave for the clipping sound.
These are what may primarily help to draw a distinction between valve type and solid state amplifiers.
The solid state amplifier supports the clean style of guitar music by amplifying on a linear level the sound produced by a guitar. It does not necessarily support the dynamic playing so essential to several musical styles. On the other hand, a valve type amplifier supports non-linear magnification of sound, thus letting you have a control over the setting of tones suited to the style of music you are playing.
The world of amplifiers is a vast one involving technology and advanced designs, and this overview has not tried to give you more than a peek into that world, and more importantly, into the basics that you need to know.