Now that we can read notes, we need some way to talk about the distance between notes, like musical meters. We use semitones. One semitone is the distance from one note to its nearest neighbor.
X:1 L:1/4 K:C C ^C w: c c♯
Here, I've noted this one-semitone distance as a melodic interval -- one note after another, like a melody. When talking about intervals in a teaching context it is more common to write them as harmonic intervals, where the notes sound together.
X:1 L:1/4 K:C [C^C]
Every number of semitones also has a specific name. For example, instead of talking about 4 semitones you can use the term major third, abbreviated M3. Here's the full table of names:
If you inspect the table for patterns, you'll find that there are 4 'perfects':
X:1 L:1/4 K:C [CC] [CF] [CG] [Cc] w: P1 P4 P5 P8
If you play these on your instrument (or a piano, if your instrument is monophonic), they tend to blend together a great deal; they are very consonant.
The other regularlity in the table is there are minor and major variants for seconds, thirds, sixths and sevenths.
X:1 L:1/4 K:C [C_D] | [CD] || [C_E] | [CE] || [C_A] [CA] || [C_B] | [CB] || w: m2 M2 m3 M3 m6 M6 m7 M7
This leaves just the tritone, which is a very important interval that I will discuss in the part about tertian chords.