The recipe
The easiest place to see Dorian is on a piano: play every white key from D up to D. That is D Dorian. The notes are identical to C major's white keys, but the scale is built around D instead of C, which moves the interval pattern from W W H W W W H (major) to W H W W W H W (Dorian). The position of that interior half step is what defines the mode.
Every mode of the major scale works this way. Start on the 2nd degree and you get Dorian; the 3rd gives Phrygian, the 4th gives Lydian, the 5th gives Mixolydian. Same seven notes, seven different centers, seven different sounds.
On the staff
What the raised 6th does
In natural minor the 6th sits a half step above the 5th and tends to fall down to it. That downward gravity is part of what gives natural minor its weight. Dorian raises the 6th by a semitone, putting it a whole step above the 5th. The pull is gone. The scale still has a minor 3rd, so it still reads as minor, but it does not feel like it is grieving.
Where you've heard it
"Scarborough Fair," in E Dorian. The raised 6th, C-sharp instead of C natural, is audible in the melody and is the reason the song does not sound funereal even though it is structurally minor. C&M cites the older "Greensleeves" as Dorian for the same reason.
Miles Davis's "So What." The A section vamps on D Dorian for sixteen bars. Bill Evans's piano voicing under that vamp, stacked fourths, is one of the most copied sounds in jazz, and it is just Dorian harmonized vertically. The link runs both ways in jazz theory: Mark Levine builds the minor 7th chord straight out of Dorian, so a D minor 7th chord and the D Dorian scale are the same seven notes read as a chord or as a scale.
"Eleanor Rigby." The Beatles in E Dorian. Beatles scholars including Alan Pollack and Walter Everett classify it this way, pointing to the raised 6th (C-sharp) in the string accompaniment.
Telling Dorian apart from the other minor modes
Dorian shares a minor 3rd with natural, harmonic, and melodic minor, and with Phrygian and Locrian. The disambiguation is in the surrounding degrees.
Versus natural minor: the raised 6th. If a minor melody hangs on the 6th and that 6th sounds bright rather than mournful, you are in Dorian. Versus Phrygian: the 2nd degree. Phrygian's 2nd is a half step above the root and reads as flamenco; Dorian's 2nd is a whole step away and stays neutral. Versus melodic minor: both raise the 6th, but melodic minor also raises the 7th, so if the leading tone pulls hard to the tonic you are hearing melodic. Dorian leaves the 7th flat, which is part of why it feels modal rather than pointed at home.
How Aubel handles it
Dorian is the first of the modes to unlock, the fifth scale overall, after the major, natural-minor, harmonic-minor, and melodic-minor foundation. By the time you meet it you have already drilled the minor 3rd against the major 3rd, which is the distinction Dorian leans on for its identity. The new thing to catch is its bright 6th, the one note that lifts it out of plain minor. Rounds play the full scale up and down from a random root, so you learn the color, not a key.