This is a loaded question, but here’s a short take that I hope is easy to understand and sufficiently device-neutral.
Distortion is any change in the input signal found at the output, aside from making the signal “bigger”. We usually talk about and measure it as harmonics (Total Harmonic Distortion or THD).
Harmonic distortions are integer multiples of the original signal frequency found on the output. The 2nd and 4th harmonics are octave musical intervals. The 3rd and 6th harmonic are a musical fifth interval and the 5th is a major third interval. The 7th harmonic is a minor seventh interval (at least that one is easy to remember). There are harmonics above the 7th, but you don’t really see them often in competently designed equipment.
Generally, tubes produce more harmonic distortion than solid state. There are reasons (single-ended operation, lack of negative feedback) and exceptions to this (push pull amps with NFB). The harmonics they produce tend to be monotonically decaying and dominated by even orders (musical octaves and fifths). This is part of what earns them the reputation of sounding ‘full’ or ‘organic.’ There is literally more musical content (even if it wasn’t intended by the original producer).
Transistor amplification generally uses negative feedback to reduce the amount of harmonic distortion in an amplifier. This is in part because transistors can (they achieve much higher gain) and partly because they must (transistors without feedback are inherently less-linear voltage gain devices). This feedback, up to a point, increases high order harmonics and beyond about 20db applied reduces all harmonic distortion. While this sounds like a free lunch, it isn’t. Lots of gain to be used in negative feedback means more stages that each contribute non-linearity and high feedback amplifiers have ugly clipping characteristics.
In the end, there is no one best device. Amps are the culmination of a series of design decisions that each have trade-offs. Unfortunately the trade-offs become the stereotypes that are bandied about among enthusiasts and you end up with ideas like ‘tube amps are full of distortion’ or ‘solid state is lifeless.’