Scale

A note an octave higher than another is vibrating exactly 2 times of frequency of the lower note. The Just (meant correct, true) system divide an octave using notes which is simple faction of the base note.  For example, the fifth is 2/3 of the frequency of the base note.  The problem is that the Just system notes may not harmonized so well among them except with the base note.  For example, the tonic and the fifth frequency coincide every a few cyclerates and thus they sound harmonized.  But the 2nd (8/9) and 6th (3/5) coincide every 27 cycles and 40 cycles respectively and thus they do not sound harmonized.

The equal temperament system uses formula instead of fraction to divide the scale.  Specifically, ratio of the frequency for every 2 consecutive semitone are the same.  Both Just and ET system use 440 Hz for A.  Comparing with Just system, the note in ET is slightly out of tune with the tonic t still close enough to sound harmonized.  However, while not all chord in Just sound harmonized, all chord in ET sounds moderately harmonized.  This is the main difference for the 2 systems

Applying Chord

Typically, chord is chosen based on the rhythmically emphasized notes, which usually the first note  in each bar.  As melody is made up by a stream of notes usually rise or fall in small tone step, not every notes in the bar would fit to the chord chosen.  Choose the chord which included most of the notes in the bar.  When transiting from one chord to next, use inverted chord could "smooth" out the transition.  For example, to transit from C to F

C: C, E, G
F: F, A, C

Each note transverse a distance of 4 tones

If we use a inverted chord for F would reduce the movement for the tones between the 2 chords

C: C, E, G
F: A, C, F

Similar we can invert C to achieve similar effect

C: E, G, C
F: F, A, C