'Solid state' with regards to amps is just a term that denotes that the circuit in the amp doesn't pass through a vacuum (tube). If your amp doesn't use tubes to pass the electrical signal through, it's solid state.
'Digial' vs 'analog' is talking about the way the audio signal is treated. With analog you have a continuous wave, with digital it is broken up into discrete steps (often for processing):
There are fully analog solid state amps (if people are being lazy, this is often what they mean when they're talking about a solid state guitar amp). There are solid state amps that are fully digital (again, if people are being lazy this is often what they mean when talking about a digital guitar amp). The Tonemaster amps are solid state digital amps - they use microprocessors and digital processing to modify your guitar signal to sound like Fender amps that are modelled and the electrical signal doesn't go through tubes.