In addition to jadarnel27's excellent answer, I think it's worth discussing diatonic intervals. A diatonic internal is one that is composed entirely of notes in a scale.
For example, in the key of C, a C major chord is made of the notes C E G. The interval between C and E is a major this. A d minor chord is spelled D F A and has a minor third between the D and the F. However, in the key of C, both intervals can be described as diatonic thirds because they are thirds in the C major scale, even though they are of different qualities.
Hence, most typical chords in a typical scale or mode can be said to be made of a diatonic third and a diatonic fifth above the chord's root.
The word "diatonic" can be used to describe other musical ideas when you're only interested in the notes in a particular key. For example, let's say you have a motif in the key of C that you want to transpose to the key of A. You can do this by transposing up a major sixth, moving all notes the same number of semitones. However, this will lead to some notes outside of the C major scale. For instance, an E would be transposed to a C#.
You could also transpose by a diatonic sixth. In this case, all notes are moved up six scale degrees in the key of C. The precise interval each note is moved will vary as a result. C will be transposed a major sixth to A and E will be transposed a minor sixth to C, but they will all stay in the key of C.