Intervals are two general categories perfect and imperfect. The perfect intervals - designated with 'P' - are unison, octave, fourth, and fifth.
P1 perfect union
P8 perfect octave
P4 perfect fourth
P5 perfect fifth
The imperfect intervals are second, third, sixth, and seventh. The imperfect intervals are of two basic qualities major 'M' and minor 'm'.
m2 minor second
M2 major second
m3 minor third
M3 major third
m6 minor sixth
M6 major sixth
m7 minor seventh
M7 major seventh
Each of the intervals can also be described as diminished 'd' or augmented 'A.' Diminished means one half step smaller than a perfect or minor interval, and augmented means one half step bigger than a perfect or major interval. Examples:
d5 diminished fifth
A4 augmented fourth
A2 augmented second (commonly found in the harmonic minor scale)
A6 augmented sixth (used with the various augemented sixth chords)
Keep in mind some interval pairs are enharmonic equivalents meaning their distances are the same, but their names are different:
P1 perfect unison = d2 diminished second
P8 perfect octave = A7 augmented seventh
m7 minor seventh = A6 augmented sixth
...some of those enharmonic equivalent names are a little strange and would be used in special notation situations.
The some other details to consider. 'Tritone' is a special name for the d5. Also, there are compound intervals which are bigger than an octave, ex. M10 major tenth = P8 + M3.
Check out the Wikipedia page for intervals for a more complete overview.