Intervals are distance between two notes. The name of the intervals (Unison, Second, Third, Fourth, Fifth, Sixth, Seventh, Octave) describe the distance letter to letter from the root note (C to D: second; C to E: third; etc). The adjectives (minor, major, perfect, diminished, augmented) are determined by the number of tones from the root note.
When you want to figure out an interval, like a minor 6th. Start by counting the number of notes (don't be afraid to count on your fingers, it is more visual that way), so starting from C: C, D, E, F, G, A. So A is the note we are looking for. Then, figure out the number of tones to know if it is minor or major. C to A has 4 1/2 tones, but we want a minor 6th which only has 4 tones, so we change or A to an Ab (A flat). Even though it is the same note as G#, the 6th is A, so we have to call the minor 6th of C is Ab. G# would be the augmented 5th.
Below is a diagram showing the different intervals up to the octave. Note that some of the intervals overlap the other like the augmented fourth and the diminished fifth which are using the same note in theory (F#/Gb), but will be used in different context.
Intervals Diagram
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