Q:

Why is the first note of a scale the most important? 

While reading and refreshing basic music theory, I've met the tonic note as related to common practice diatonic scales. 
From introductory resources, the usual explanation is simple enough.  The first (the lowest) note of the scale is the tonic note; the most important note of the scale, the centre around which a melody tends to be oriented, and where it feels resolved. 
However, I've never seen any further justification of why the first note of the scale is the tonic.  It's just usually stated for the reader to accept. 
Given my limited knowledge and reading, I can think of two possible explanations (that may well be far off the mark):

The choice is somewhat arbitrary.  When we construct a scale on a note, there's a convention that the first note will be most important in the composition. 
A composer then goes about deliberately following this rule and establishing that first note as most important.  And, maybe, through continued exposure to music constructed this way, listeners expect certain scales to tend to particular notes. 
There is actually a fundamental underlying physical or perceptual reason why the first note is the most important.  It's not just an arbitrary choice. 
I can certainly hear that the tonic behaves this way, e.g. when a scale is played, it feels resolved when you reach the starting note one octave up.  Perhaps this is due to particular interval relationships, or some other effect? ( Or maybe this is an effect of familiarity, as mentioned above in 1...)

A:

The first note of a scale is the tonic (or root of the key) by definition.  We build scales off the tonic; we do not define the tonic based on the scale.  
The tonic itself comes from the key the piece is based in.  The key is defined somewhat by the accidentals, but also by the harmony of the piece.  You'll run into music theory words like cadence here.  You hear this theory as the resolution that you mention in your answer. 
Regarding the eighth note sounding resolved, it might be worth mentioning that octaves have a specific frequency relationship.  The frequency is exactly double the note an octave below.  Our ears hear this as the same pitch class.  So there are some physical reasons at play. 