What is "touch-sensitive"?
"Touch-sensitive" is an adjective that guitarists and manufacturers alike use to describe amplifiers. Like so many adjectives used by guitarists and manufacturers, its meaning seems to mutate depending upon context. In actuality, touch-sensitivity is not an absolute quantity, but rather a spectrum of behaviors.
In its broadest sense, touch-sensitivity is some change in an amplifier's behavior - other than getting louder or softer - as the player changes picking intensity or the guitar's volume control. With the exception of a few specialty amps (mostly designed for jazz or acoustic guitars), almost all guitar amps have some degree of touch sensitivity. The devil, as they say, is in the details.
At its most basic, touch-sensitivity is an increase in distortion with an increase in volume or picking intensity. It doesn't matter whether you're cranking up a vintage amp to get that distortion, or using the gain on a master-volume amp. Distortion is distortion, and you can virtually always clean up the distortion by reducing your picking intensity or guitar volume.
Of course, there are nuances. In the world of musical instruments, there are always nuances...
Distortion can come from any of the amplifier stages. In master-volume amps, most of the distortion comes from the preamp. The character of preamp distortion varies with touch in a manner that depends upon how the preamp is "voiced". The number of stages, frequency shaping, type and location of the tone stack, gain staging (i.e. the gain of each stage, and any gain reduction between stages), all affect touch sensitivity. As a rule of thumb, the more gain stages you have, the lighter the touch needed to "clean up" the sound. At some point, depending upon your playing style, too much gain works against touch sensitivity. You can always roll back the gain control on the amp, but those extra stages and their associated tone shaping will color the sound of your guitar. That coloration may be for better or worse, depending on your preferences.
The output stage of the amp can also contribute its own distortion. This tends to happen in vintage circuit designs, but may also be part of the sound of a master-volume amp when the gain is run low and the master high. Output-stage distortion is commonly referred to as "power tube" distortion, despite the contribution and effect many components. Every component of the output stage (which includes the phase inverter or driver) contributes to the voicing of the stage. There's also may be an interaction between the power stage and the power supply. The output stage/power supply interaction, the behavior of the output transformer, and other factors combine to make output stage distortion more "complex" than preamp distortion.
A speaker can also contribute distortion in a few different ways. The motion of the voice coil may (depending, of course, on the design of the speaker) become nonlinear at high powers. Alnico-magnet speakers become nonlinear at high volumes because the magnetizing force of the voice coil partially (and temporarily) reduces the field of the magnet itself. Finally, the paper cone fails to move as a unit at high amplitudes and certain frequencies. The resulting multimodal vibration introduces distortions and colorations (some desirable, others not) of the sound coming from the speaker. As you might imagine, speakers are much more linear at lower volumes, which is one reason that power attenuators change the sound of your amp so noticeably when you cut too much of the room volume.
So, what makes for a "good" touch-sensitive amp? It depends on what you're looking for. I like an amp that I can clean up using the guitar's volume without the sound getting muddy. I pick with my fingers and use fairly light strings. I use vintage-output pickups set fairly low for clarity. And my picking intensity can range from very light to what I'd consider "normal". I like to get a horn-like distortion when I dig in - I don't really care for smooth, saturated distortion. I learned - the hard way - that Boogies have way too much gain for my playing style. As part of the same lesson, I learned that Boogies don't really get into output-stage distortion. I've played a lot of Fender blackface- and silverface-era amps, and almost always felt like I was killing either the guitar or my ears by the time they gave up the goods. I eventually found that tweed- and brown-era Fender designs worked the best for me.
Combine a simple signal path with a design that introduces distortions that change in character (not just intensity) as a function of volume: that, for me, is the recipe for a good touch-sensitive amp. Low-level inputs give crystalline cleans. Increasing levels "warm up" the tone with subtle distortions. Further increases make the distortion apparent, introducing a noticeable "edge" that fades as the note decays. Dig in even harder, and the tone starts to compress, but doesn't flatten out - there's still more room. The speakers are starting to growl now. Dig in even harder, and the notes bloom. That's my idea of a good touch-sensitive amp, and I've found it in several different amps. It's not a holy-grail kind of thing - it's mostly a function of the amp's design and (to a lesser extent) some of the critical components. Once you know what to look for, you'll be able to find it in a lot of amps.
Finally, remember that touch-sensitivity is a function of amp design, but the amp has to be matched to the player and the guitar. Keep that in mind the next time you hear about some holy-grail touch-sensitive amp.