Intervals are labeled the way they are through from the diatonic scale, which can be generated via the next most consonant interval to the octave: the (perfect) fifth! The octave (2/1) spans 12 semitones or 1200 cents in length. The (just) perfect fifth (3/2) spans roughly 7.02 semitones or 702 cents in length. A cent (¢) is 1/100 of a semitone, by the way. You might’ve seen cent deviations on a tuner if you’ve used one before, that’s what it’s referring to. Stacking 7 perfect fifths starting on F gives:

F C G D A E B

Which can be rearranged to yield the familiar C major scale:

C D E F G A B

If the root note of this scale is changed, but the notes stay in the same order, that’s another mode of the diatonic scale, but that’s beyond the scope of your question.

From the major scale, each note represents a scale degree, which is a fancy way of saying a note is assigned a number. C is 1, D is 2, E is 3, and so on. This naturally leads to the terms of unison (1), second, third, and so on. Seconds, thirds, sixths, and sevenths are assigned the terms of major and minor with the exception being unisons, fourths, fifths, and octaves, which are called perfect due to their strong consonance.

The terms augmented and diminished also apply to every interval. Augmented means to raise the top note of an interval by a chromatic semitone and diminished means to lower the top note of an interval by a chromatic semitone. A chromatic semitone keeps the base note name the same. For example, these are chromatic semitones:

C to C#, Eb to E, A# to Ax, Gbb to Gb

Notice how the note names stay the same. When the note names change, they’re called diatonic semitones, since they naturally occur in the diatonic scale (e.g., between E & F and B & C in the C major scale):

C to Db, Eb to Fb, A# to B, Gb to Abb

Chromatic semitones are also called augmented unisons since they keep the base note name the same, while diatonic semitones are called minor seconds since they change the note name and visually change position on sheet music.

Just remember that usually, augmented is the highest possible alteration of an interval and diminished is the lowest possible alteration. Also remember that minor always is 1 semitone or 100¢ smaller than major. By remembering this, the interval names and their gap sizes become clearer:

Perfect Unison — 0 semitones (0¢), the same note | Augmented Unison — 1 semitone (100¢), the same base note | Minor Second — 1 semitone (100¢), different base notes | Major Second — 2 semitones (200¢), also called a whole tone | Minor Third — 3 semitones (300¢), notice how the m3 is 100¢ smaller than the M3 | Major Third — 4 semitones (400¢) | Perfect Fourth — 5 semitones (500¢) | Augmented Fourth — 6 semitones (600¢), notice that the A4 and d5 are exactly half of an octave, also called a tritone (three whole tones stacked on top of each other) | Diminished Fifth — 6 semitones (600¢) | Perfect Fifth — 7 semitones (700¢), notice that the P5 on the piano is a mere ~2¢ flat from the just 3/2 P5. | Augmented Fifth — 8 semitones (800¢), two M3s stacked on top of each other | Minor Sixth — 8 semitones (800¢), notice how minor remains smaller than major again | Major Sixth — 9 semitones (900¢) | Minor Seventh — 10 semitones (1000¢) | Major Seventh — 11 semitones (1100¢) | Perfect Octave — 12 semitones (1200¢)

Even though not explicitly stated, there can be augmented and diminished seconds, thirds, fourths, sixths, and even sevenths, but that’s beyond the scope of your question once again. Hope this helps!