The pattern, and why the half steps matter
A major scale is seven different letter names plus the octave on top, arranged in the fixed step pattern W W H W W W H. The two half steps are the landmarks. They sit between the 3rd and 4th degrees and between the 7th and 8th, and they never move. If you can hear where the half steps fall, you can tell a major scale from a mode that looks similar on paper but slides one of those half steps to a different place. Kostka and Payne describe the scale as two identical four-note tetrachords a whole step apart, each running whole, whole, half, which is another way to see why the half steps land where they do.
The piano makes the pattern visible: play every white key from C up to the next C and you have played C major, half steps and all (the white-key half steps are exactly E to F and B to C). Start that same white-key run on any other note and the half steps land in the wrong places. That is what a mode is, and why the other white-key scales sound different even though they use identical notes.
On the staff
The two degrees that carry the sound
Not every note in the scale is equally distinctive. The 3rd degree, the major 3rd, four semitones up, is the one that makes the scale major rather than minor; lower it a semitone and the whole thing tips into minor. The 7th degree is the other character note. It sits a half step under the octave and pulls upward into it, which is why it is called the leading tone. That upward pull is most of what makes a major scale feel like it is heading somewhere and then arriving.
For ear training, that means you are rarely listening to all seven notes equally. You are listening for a bright 3rd and a leading tone that snaps into the octave. Get those two, and the scale announces itself.
Where you've heard it
"Do-Re-Mi" from The Sound of Music is the major scale set to lyrics, one degree per syllable, ascending, close to a sung diagram of the scale. "Joy to the World" opens with the same scale running the other way, a full descent from the octave back down to the tonic. Holding those two in your head gives you the rising and falling shape, which is exactly what the app plays.
How Aubel handles it
Major is one of the two scales the Scales game starts with, alongside natural minor. That pairing is deliberate: the first thing the game asks you to do is hear major against minor, the major-3rd-versus-minor-3rd distinction that everything else builds on. Each round plays the full scale ascending and descending from a random root, so you are naming the pattern, not a particular key. Once that pair is solid, harmonic minor unlocks next, then melodic minor, then the modes.