When referring to amplifiers the favorite catch-phrase of audiophiles
seems to be "clean power". As an ignorant novice I would like to know
what is meant by "clean power" and how do I achieve it.

There's no such thing. Power is power. What's usually meant is low
distortion and noise. It's easily attainable simply by not driving your amp
into clipping. That means use sufficient amplification for your needs.

Also, I once
heard something about voltage matching an amplifier with a head unit
but if I set my amp to 4v (same as the preouts on my head unit) I can
barely hear my subs. Do I need to add a bigger amp to get the benefit
of voltage matching? Will this theoretically provide "cleaner power"?

No. NEVER match your amplifier's gain setting to the nominal rating on your
head unit. It doesn't work because a) the output voltage of the HU doesn't
always meet the nominal rating, especially when listening to quieter
passages; b) filtering out higher frequencies with the LPF reduces voltage.
Go strictly by ear when setting gains. There's a tutorial at
installer.com's tech pages.