The mnemonics
"When the Saints Go Marching In" opens with a major 3rd up: the "Oh" to "when" leap on the first two words. Kumbaya does the same thing on the first syllable to the second. Hush Little Baby on "Hush" to "lit-". The piano sample above plays C and E.
Flatten the second note by a semitone (E to E♭) and the interval becomes a minor 3rd. That single semitone is the entire difference between a major chord and a minor chord. It's why most people can hear "happy" and "sad" in a chord before they've studied any theory at all.
What the major 3rd tells you
Almost every triad you'll encounter uses a 3rd as its second note. The other two notes (root and 5th) tend to stay constant; the 3rd is what changes. Major triads have a major 3rd. Minor triads have a minor 3rd. Major 7 and dominant 7 chords sit on a major 3rd. Minor 7 and half-diminished sit on a minor 3rd. If you can hear that distinction reliably, half of chord ear training is already done.
Where it gets confusing
The major 3rd has two close neighbors that catch most people for the first few weeks: the minor 3rd (one semitone narrower) and the perfect 4th (one semitone wider).
Versus the minor 3rd: imagine continuing the interval up into a full triad. If the resulting chord sounds bright, you heard a major 3rd. If humming the triad feels heavy, you heard the minor.
Versus the perfect 4th: the 4th feels open and unresolved. The major 3rd feels like it has arrived somewhere. Hum the second note for an extra beat. If it sits comfortably, it's the 3rd. If it wants to go up, it's the 4th.
How Aubel handles it
The major 3rd isn't in the absolute starting pool. It unlocks after you're consistent on the perfect intervals (unison, 4th, 5th, octave). For the first three sessions after unlock, Aubel weights it a little heavier than its share, which is enough exposure for the sound to settle in. After that it rotates in normally with the rest of the pool.