Actually this is how it goes:
1. The signal from your electric guitar is pretty weak, and is insufficient to drive the speakers significantly and produce audible sound.
2. So you need to amplify (=increase) the signal.
3. This is done via amplifiers
4. Amplifier has two sections - pre-amp and power amp.
5. Pre-amp's purpose is to bring the signal (from various sources) to a standard level.
6. Power amp's purpose is to now amplify this signal to drive the speakers.
7. In many cases - your pre-amp can be either valve/tube or solid state analog or solid state digital.
8. Further the power amps are either valve/tube or solid state analog.
Now when you call something as a tube amp - it usually means that your pre amp and power amp sections both are driven by valves.
When you call something as Solid state amp - usually it means that your pre-amp and power amp sections both are driven by solid state analog devices
When you call something as digital amp - it usually means that your pre-amp section is solid state digital, and power amp section is solid state analog.
So what is the audible difference?
Unfortunately only your ears (and feedback from your fingers) can detect it. No one can actually describe it.
You asked for advantage-disadvantages:
1. Tube amp sounds much better to many musicians because they believe it is more organic sounding etc. Personally I feel its because they have a particular tone in their mind which harks back to the days when there were only tube amps. So now everyone wants that tone.
2. Solid state amps are more robust mechanically. Amps like Tech21 can sound pretty close to the tube amps, or different - but with similar characteristic.
3. Digi amp have modelling sections - so one amp can try to sound like a 100 of different amp (just like a digital processor like POD). But many ppl don't prefer the tone.
You donít get what you deserve, you get what you negotiate.