Intervals are numbered based on their position in the scale.

Here is the C major Scale
C  D  E  F  G  A  B  C
1   2  3  4   5   6  7   8

Using numbers to refer to tones is used extensively in music.

D is the second and is called the Major 2nd if it was Db a half a step lower it would the Minor Second.

All the numbers list above are Major except for the 4th and 5th which are called Perfect. The second C or 8 is the Octave.

The numbers used are called the scale degrees, so D is the second scale degree of the C Major scale.

Unison
The same tone C to C.

Major 2nd
In C Major Scale
The second interval would be from the root note, C in this case, to D.
This is also referred to as the Major 2nd interval.

Minor 2nd
The interval between C and Db would be the Minor 2nd since D is lowered a half step.

Major 3rd
From C to E.

Minor 3rd
From C to Eb, again lower the 3rd by half step and it is a Minor 3rd.

Perfect Intervals
Perfect intervals cannot be Major or Minor, the term perfect is because the note sounds like the root.
Perfect intervals are Unison, Perfect 4 (P4), Perfect 5th (P5) and the Octave.

Diminished Intervals
If you lower a Perfect interval by half step they become Diminished Intervals.

C to G is a Perfect 5th but C to Gb is a Diminished 5th.

Augmented Intervals
If you raise a Major or a Perfect interval by a half step they become Augmented Intervals.

C to G is a Perfect 5th but C to G# is a Diminished 5th.
C to D is a Major 2nd, but C to D# is a Diminished 2nd.

Intervals can be abbreviated with capital M meaning Major, small m meaning Minor and P for Perfect. This creates the shorthand of M2,m2,M3,m3,P4,P5,M6,m6,M7 and ,m7.

Harmonic or Melodic

If the notes are played simultaneously they are Harmonic intervals.
If they are play consecutively they are Melodic Intervals.