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Analogue tape (magnetic tape) has a property that causes it to saturate when overloaded. This is a form of distortion, however, it tends to sound 'pleasant' to the human ear/brain. Digital distortion, on the other hand, never sounds 'pleasant' but can be what some people want to hear (e.g. those into 'glitch' genres/artists such as Aphex Twin and the like).
Distortion usually occurs several dB before the clipping point of most (analogue and digital) gear. For most digital recording/playback devices '0dB' is the 'clipping point' and, bearing my previous paragraph in mind, the point where signals start to distort is below this point, i.e. -ndB. This only begins to tell the story, and to know/understand the full story you'll need to know what decibels are and how they're measured.
To that end, allow me to point you in the direction of this reference. Now if I add to that, that 0dBFS can be anywhere from +18 to +24dBu (depending upon the manufacturer and which standard they chose to follow), you can see how easy it is to introduce distortion into a digital mix without realizing it (at least for someone who didn't know this beforehand).
The best way to control peaks is in the performance. Failing that, you can apply compression to get the peaks in line (although track automation is my preferred method, sometimes followed by an appropriate amount of compression to help 'gel' the signals).
Loudness isn't everything and compression (and limiting) is designed to squash the dynamic range of the signal(s) passed through the compressor (limiter), which in turn (when over used/misused) kicks the life out of the part it's applied to. Fine if that's what you want, but often it's not what's wanted at all.
It's also worth investigating the frequency ranges that the instruments in your compositions occupy. If two (or more) instruments have frequency ranges that overlap, e.g. kick drum and bass synth/guitar, you can avoid some muddiness by reducing the level of one using an EQ cut in the conflicting range and/or applying 'ducking' (compressor 'pulls down' the level on one instrument whenever the second instrument sounds on top of it), although outside of voice over and kick/bass, EQ cutting is usually preferred to ducking.