Major third interval

Major third in musical notationThe major third interval consists of two notes with two whole steps distance. For example, C to E note will result in this musical interval.

The major third interval is abbreviated M3 whereas the minor third is abbreviated m3. An alternate spelling of major third is diminished fourth.
Keyboard major third interval 
A major third with C to E on the keyboard.

Listen to major third interval (C-E):

This interval is rare in scales and known as a quadra step. It can be found in some exotic scales such as the Balinese and Chinese.

The major third has on the other hand a key role in chords. It is the third that decides if a chord has a major or minor quality. Chords with a major third interval between the first and second notes are in general major chords. For example, C consisting of the notes C, E, G and Cmaj7 consisting of the notes C, E, G, B.

Major thirds result in a pleasant sound and are often part of the harmonic in compositions. Below is a simple example with only major thirds intervals (some are inverted):
Piano notes example with major thirds 

Ascending major third intervals

A list with ascending major third intervals:

Descending major third intervals

A list with descending major third intervals:

Enharmonic notes such as Cb and Fb are leaved out.