Boosting the input to the amp (or just cranking up the volume) starts to overdrive the preamplifier and power amplifier and produces a somewhat distorted sound that is fuller and has more sustain.
Overdriving a tube amplifier will produce a larger, "flattened" signal with (what most players consider) warm overtones. By any definition, this is a distorted signal - the output is different from the input.
Transistor (solid-state) amplifiers distort in a different manner - the top of the signal is "clipped" producing a flat signal with many odd harmonics. Adding an infinite number of odd harmonics to a pure audio signal produces a square wave, which is considered to be a harsh sound. This is why early transistor amplifiers weren't liked by many guitar players.
Newer solid-state amplifiers use different technologies to produce a warmer, tube-like sound.
To answer your original question, overdriving an amp is normally not dangerous to your equipment. I have been overdriving my 1965 Fender Super Reverb since 1966 - no failures from overdriving. If you look at the channel-switching amps available today, they are designed to be overdriven. The "dirty" channel uses the amplified signal from the clean channel to overdrive the amp.
One caution - the combination of power and distortion can take a toll on speakers. A large square-wave type signal can overdrive and damage the speakers. However, this is a risk that we all take when playing loud. I have never blown a speaker in my Fender, and it used to run on 10 at all my gigs.
