An augmented third is an interval of five semitones.
The interval between C and F, for example, is an augmented third as it contains five semitones.
In a lot of common scales, such as the standard major scale and the natural minor scale, the interval between the root on the scale and the third note on the scale is either three semitones or four semitones. These more common intervals are called a minor third and a major third respectively. The augmented third is "augmented" as it is one semitone larger than the larger of the two more common intervals (the major third). The augmented third is a "third" if it occurs between the first and the third note on a scale (or more generally between two notes with one note in between).
In principle, an augmented third is the same as a perfect fourth. Both intervals contain five semitones. An augmented third through would refer to the interval between the root on a scale and the third note on the scale (or more generally between two notes with a note in between) whereas a perfect fourth would refer to the interval between the root on a scale and the fourth note on the scale (or more generally between two notes with two notes in between).
Even though the augmented third is not as common as the minor and the major third, there are scales that use this interval. The enigmatic scale in C, for example, is C, Db, E, F#, G#, A#, B, and the interval between Db and F# is equal to five semitones.
Intervals on the chromatic scale, Intervals on the chromatic scale (index)