Just want to add some possible help in translating things into lay terms.
Keep in mind that there’s the engineer’s definition of “solid state” and there’s the phrase “solid state” that is used in marketing. Mangetout and Balthisar have probably given the shortest, most meaningful definition for consumer devices. Zenster’s was more to my liking, but I can see where some of his description may have strayed into the esoteric.
I think there’s a bit of confusion because “solid state” has evolved over time from meaning “there are transistors in here” into “we’ve gotten rid of (almost) all mechanical components.”
The basic vacuum tube or transistor does one thing: it will allow current to flow from connector A to connector B depending on whether or not there’s current at connector C. The vacuum tube does this by bending an electron beam sideways by passing it through either an electromagnet or a capacitor. A capacitor is just two plates sitting next to each other maintaining a voltage difference. Needless to say, bending electron beams requires delicate and precise manufacturing, and they’re big. (Just trying to remember back into the 60’s, here, but I think the smallest one I saw was about an inch and a half long and about 3/4" in diameter).
Transistors, on the other hand, use the voltage on “C” to change the conductivity of a “semiconductor.” With sufficient voltage, a semiconductor will conduct; without that voltage, it won’t conduct. As it turns out, we (as a species) couldn’t figure out that semiconductors existed until we figured out some basic quantum mechanics. (I think it’s cool that we’ve got so many real-life examples of "better living through quantum mechanics." However, vacuum tubes are miniature particle accelerators, which is also pretty cool.)
Vacuum tubes have a linear response because the amount of force acting on each electron is directly proportional to the voltage difference between the plates I mentioned above. (I’m lying, of course – quantum mechanics gets involved here as well, as it always does, but when you add up the cumulative effects of a very large number of electrons doing something versus not doing something, it ends up looking like there’s a smoothly-varying behavior). Transistors have a more difficult time exhibiting linear behavior because it’s not in their nature – they are approximately linear over a very small voltage range, and (IIRC) it’s dependent largely on our inability to control the purity of the substrate (e.g., silicon), the doping process that Zenster referred to (basically, jamming atoms of the doping substance into the substrate) and the size of the metal connectors (basically, spraying metal onto the top of the substrate).
If you care about the “fidelity” of the signal over ranges of voltages and frequencies, you need (among other things) a good linear amplifier. At first, we used vacuum tubes because they were all we had. Audiophiles tended to like tube amplifiers even after some good high-power-and-fairly-linear transistors came on the market because of the linear response of the tubes. I’m not an audiophile, so I’m not sure if they still feel that way. (They also developed severe facial twitches when so many of us regular people adopted the notion that Compact Disks sounded just fine, but I’m not sure where they stand on that nowadays).
If all you care about is “on or off,” a vacuum tube is overkill. You can use a relay (a physical switch that’s set or reset pulling a contact away from another contact using an electromagnet) or a transistor. The good thing about a relay is that it can handle lots of current – the signal that you send to the electromagnet is completely separate from the current flowing through the contacts. So we still use relays in televisions for the main current going into the Cathode Ray Tube (CRT), and in receivers for the main current going into the amplfier. The bad thing about relays is that they’re slow – we’re actually waiting for a physical contact to be moved through space, using an electromagnet and a spring. A vacuum tube can be switched faster, because we’re only moving a few gazillion electrons. However, we’re still waiting for a voltage to accumulate on the plates (connected to C), and we’re still waiting for the electrons to traverse the distance between A and B. A transistor can be switched even faster, because although we’re still waiting for a voltage to accumulate (on a plate connected to C), it’s only got to accumulate in a very small area, and it’s not bending an electron beam, it’s just changing the energy state of a little area of the semiconductor. (Don’t make me try to explain that last part – I’d only embarrass myself).
Suffice to say, if you are trying to put a few billion switches together and they have to be able to switch on and off really quickly, it’ll take quite a bit of effort, and you’ll end up with something the size of a planet (like, say, Forbiden Planet).
BTW: Some modern radars use solid state circuits to maintain a coherent, extremely high frequency, waveform. That allows them to be used as pulsed Doppler radars, in which we measure the frequency shift between a reflected, returned pulse and the original transmitted pulse. Earlier radars relied on “Gridded Traveling Wave Tubes” to maintain their coherency. Gridded Traveling Wave Tubes are very similar to, um, 1920s Style Death Rays.