The simple answer is that a tube amp uses one or more vacuum tubes to amplify the signal, while a solid-state amp uses solid-state electronics (diodes, transistors, etc.) to amplify the signal. On paper and in theory these two implementations should yield identical result, but in actuality the difference is usually noticeable.
But the simple answer fails to answer to the complexity of the issue. Many amps are not simply tube or solid-state, but mixes of both kinds, called "hybrids." This usually means that they have a tube preamp stage, employing vacuum tubes in the tone shaping circuitry, but use solid-state circuitry for the power section. The hybrids are closer to full tube amps in response and tonal warmth, but purists will still find a difference between the two. Tube amps are generally more expensive in initial cost and to operate (because you need to replace the tubes occasionally), and solid-state amps are generally less delicate and more reliable. Many players, however, feel that tube amps yield a warmer, more musical tone and better distortion.
Yet another wrinkle is tube emulation circuitry. Many amps and preamps have sophisticated circuits designed to act like tubes, and as in all things, some are better than others. The newest develop in amps are the modeling amps, which not only emulate the tone and response of tubes, but of specific tube amps. These are in general pretty exciting amps, but again, some are better than others at getting specific models, and in maintaining the sounds through a range of volume levels.
Another point to make about tube amps is that bigger is not always better. You get the distinctive tube sound most when the amp is cranked up enough that the tubes are saturated or nearly saturated. For this reason, it is often better to choose a lower wattage amp over a higher wattage amp, depending on how and where you play. By the time you crank up your 60 watt amp enough to saturate the tubes to get just the right level of distortion, you could be blowing your audience out the back door. It might have been better to choose a 20W amp that lets you get your saturated tone without the ear-killing decibels.