Overdriving an amp is a normal part of today's (and yesterday's) sounds. The early rock-and-rollers found that when they cranked up their old Fender amps, the signal started to distort and they said, "Hey, that's pretty cool!"
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.