Diatonic Scales
A diatonic scale is not a specific scale, but rather a way a scale (or a chord for that matter) is constructed. Diatonic scales are constructed from a mix of whole and half steps (dia means two in Latin), in a contrary way to chromatic scales which are constructed only by half steps.
Below, a chromatic scale (C Chromatic).
Below, a diatonic scale (C Major).
Notes, scales and chords
There are both diatonic scales and chords. When chords coincide with the the notes in a scale, they can be said to be diatonic. The chord progression C - F - Am - G is diatonic in the way all chords only include notes from the C Major scale.
The term diatonic is going back to the ancient Greece, where musicians separated octaves into intervals with names that all begun on dia-.
A diatonic scale is based on seven whole steps of perfect fifths: C - G - D - A - E - B - F. In modern Western music a scale is referred to as diatonic if it is based on five of whole steps together with two half steps.
The Major Scale has this formula: Whole, Whole, Half, Whole, Whole, Whole, Half. The same is true, although the order shifts, for the formula of the Natural Minor: Whole, Half, Whole, Whole, Half, Whole, Whole.
Many scales are diatonic, such as Major, Minor (the Harmonic minor is an exception) and modal scales. Examples of non-diatonic scales are pentatonic, octatonic and whole-tone scales.
The thing to remember is that diatonic is a central term in music theory that can describe to scales and chords. Diatonic can relate to scales that consist of seven tones that can be ordered in some specific way, for example, by five whole tones and two half steps. It can also describe chord which are related to the same major or minor key. For example, the chords C, Dm, Em, F, G, Am and Bdim are diatonic chords based on the C Major Scale. See Harmonizing major scales into chords for more examples.
Sources:
Music and Twentieth-Century Tonality: Harmonic Progression Based on Modality and the Interval Cycles (Routledge Studies in Music Theory) by Paolo Susann and Elliott Antokoletz
Wikipedia.org, https://en.wikipedia.org/wiki/Diatonic_scale