What is in it
A dominant 7 is four notes: a major triad (root, major 3rd, perfect 5th) with a minor 7th added above. From C that is C, E, G, B flat. The major triad makes the bottom of the chord sound bright and stable, but the minor 7th on top changes everything, because it forms a tritone with the 3rd.
On the staff
The tritone is the engine
Take the 3rd and the 7th of a C7, the E and the B flat, and you have a tritone, the most restless interval there is. Both notes want to move: the E pulls up toward F, the B flat pulls down toward A. The chord they point to is F, a perfect 5th below the root. This is why the dominant 7 is the great resolving chord. It is not stable in itself; it is a tension built to discharge into a tonic.
Where you hear it
In a major key the dominant 7 is the chord on the 5th degree, V7, the one that leads back to I, and it closes most classical cadences. It is also the whole sound of the blues, where the I, IV, and V chords are typically all dominant 7ths, a usage that ignores the classical urge to resolve and simply enjoys the colour. If a single chord could be said to drive Western harmony, it is this one.
Jazz puts it at the centre of its most common progression. Mark Levine treats the ii-V-I as the bedrock of the music: a minor 7th, then this dominant 7th, then a major 7th. The 7th of each chord falls a half step to become the 3rd of the next, so the dominant is the link that holds the tension and then hands it down into the tonic.
How Aubel handles it
Dominant 7 unlocks fifth, the first of the five 7th chords. Your ear already has the major triad locked in from the starting pool, so the new thing to catch is the minor 7th sitting on top of it and the slight edge the tritone gives the chord. The cue is a bright triad that nonetheless sounds like it wants to go somewhere. Major 7 unlocks next, the same triad with the 7th raised a semitone.