An interval is the distance between two notes. So, if we use the C major scale as an example (C,D,E,F,G,A,B) then the note E is a third above C. The note G is a third above E, and the B is a third above G.

The dorian example shows a flattened third and seventh, meaning that the third and seventh notes of the scale are dropped by a semi-tone (1 fret) as compared to the major scale...

Interval recognition is useful, but it's actually easier to work out using arpeggios than scales or chords. If you wanted to find out what a major ninth above A was, playing a major 9th chord would help you, but it would be easier to look at the arpeggio. (and in this poor example, it would actually be easier to think that a 9th is the same as a 2nd, so a 9th above A would be B... :D )