Could anybody explain to me what is affected by Just Intonation? The melodic and harmonic intervals or the scale degrees and distance from the tonic?
Just Intonation is a tuning system; that is, it defines the tuning of a scale. We commonly use equal temperament, which is a compromise (or temperament) that allows us to play in all keys with a limited number of notes (without constant retuning).
Just Intonation defines the actual size of the intervals, or the tuning of the notes with respect to the tonic (as does any temperament). If you look at the Wikipedia article for Equal Temperament, there's a good comparison to Just Intonation.
I use this practically when I'm playing in a concert band. We can dynamically change the tuning of our instruments using our mouth positioning (embouchure), tuning slides, garden variety slides (i.e. trombones), and so on. When I know I've got a third in a chord, I'll play it slightly flat (in equal temperament). This results in just intonation. If you split all the parts up and used a equally tempered tuner, you'd think our thirds were flat, and our fifths were slightly sharp. That's the effect of using just intonation. It changes the tuning system.
It's worth noting that none of the note names or scales change. There isn't a notation that tells us to play with just intonation, but we tend to do so if it's possible, because it sounds better.
To add to endorph's answer:
If you're used to 12-equal temperament (the octave divided into 12 equal semitones), then playing in just intonation entails making distinctions you hadn't made before: there are situations where two notes would be played at the same pitch in 12-equal but at different pitches in just intonation.
In 12-equal, there is a 12-step circle of fifths. But in just intonation, a sequence of downward fifths or upward fourths never gets back to its starting point.
This might seem obvious -- after all, even if we play e.g. G# and A flat at the same pitch, the note names are different, which gives a clue that they are not inherently identical. But there's another issue which means that even notes with the same name can sometimes need to be played as different pitches.
Start, say, at C, and go up four just perfect fifths. This takes you to G, D, A and E. But a major third above C is a little lower than E -- it is lower by a syntonic comma. (Sorry about the additional jargon on that web page, but near the bottom of it is some notation on two staves which should, I hope, make things clearer.)
If you are playing in just intonation, you will need to work out how many syntonic commas to lower or raise notes so that your thirds are in tune: a major third needs to be narrowed by a comma (e.g. plain C to lowered E) and a minor third needs to be widened by a comma (e.g. lowered E to plain G).
A thing to be aware of is that some chord progressions that are very common in tonal harmony cause the pitch to drift downwards by a syntonic comma. For example, suppose that the key is C major, and there's the chord sequence C major, F major, d minor, G major, C major. The initial C is plain (not lowered); so's the F of the F major chord. So that F major chord's third, A, is lowered by one comma. So in the d minor chord, the A is lowered (and the F is plain), so the D is lowered, too. So the following G and C are lowered, too. The phenomenon's known as (syntonic) comma drift.
In Syntonic Tuning: A Sixteenth-Century Composer's Soundscape (Music Theory Online: the Online Journal for the Society for Music Theory, Volume 10, Number 1, February 2004), Roger Wibberley cites a passage from Josquin des Prez's Ave Maria Virgo Serena where, in the course of 8 bars (45-52) the pitch drops by five syntonic commas! (Wibberley's example 1 is b.44-53.)