An 'interval' is the distance between any two pitches.
A melodic interval is the distance between two successive pitches and a harmonic interval is one between two simultaneous pitches.
Intervals are named according to the number of letter names they encompass and the quality or size of the interval. In staff notation, letter names correspond with lines and spaces, so you could also use the number of lines or spaces included when naming an interval.
For example, the intervals from C to E and D to F are thirds because they cover three letters (C, D, E and D, E, F); E upward to A is a fourth because it covers 4 letters, or four lines/spaces: E, F, G, A.
The quality of an interval is determined by the number of half steps it contains.
Seconds, thirds, sixths, and sevenths can be either major or minor. A third with 4 half steps is major (like C-E), and one with only 3 half steps (like D-F) is minor.
Unisons, fourths, fifths, and octaves are not major or minor: they are either perfect, diminished, or augmented. For example, a fourth with 5 half steps is perfect (like E upward to A). But F rising to B is augmented, because it is one half-step larger.
Seconds, thirds, sixths, and sevenths can also be augmented or diminished: a third one half step smaller than minor is diminished (like D to Fb). A second one half step larger than major is augmented (like C to D#).
You'll have noticed that some intervals are enharmonically equivalent to others. On a piano you use the same keys to play both C-E# and C-F, so the augmented third is enharmonically equivalent to the perfect fourth.
The traditions of melodic writing in counterpoint generally discourage writing any diminished or augmented melodic intervals, which are thought to be awkward. This is true even if the interval is enharmonically equivalent to a better interval.
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