The tritone, and the interval that cannot sit still
Three whole tones, the interval that splits the octave exactly in half; the only symmetrical one, the one that makes every dominant ask to resolve, and the one that sounds tense even to someone who never studied.
Play C and F sharp together. The ear asks for a continuation: it is the only interval in the scale that sounds like the middle of a sentence. Three whole tones, six semitones, and a name that says exactly that: tritone. It is the engine of tonal harmony, and it is worth knowing as an interval before meeting it inside a chord.
Half the octave
The octave has twelve semitones. The tritone has six: it splits the octave exactly in half. It is the only interval with that property, and the property has a strange consequence: the inversion of a tritone is another tritone. Invert a third and a sixth comes out; invert a fifth and a fourth comes out; invert a tritone, from C-to-F♯ into F♯-to-C, and the same six semitones come out.
So it has no "direction" of its own: nothing in it says which of the two notes is the lower. That is what makes it unstable, and it is also what makes it useful.
Two names, two resolutions
Like every six-semitone interval, the tritone has two spellings, and the spelling decides where it goes. C to F sharp is an augmented fourth: the fourth stretched by a semitone. C to G flat is a diminished fifth: the fifth shrunk by a semitone.
The sound is the same; the tendency is not. The augmented fourth opens: the two notes move apart, to B and G. The diminished fifth closes: the two notes move together, to D flat and F. Whoever writes chooses the name by the destination, and whoever reads knows the destination by the name.
Where it lives in the scale
A major scale has exactly one tritone, between the fourth and seventh degrees. In C major, F and B.
Those two degrees are precisely the ones that form semitones with their neighbours: F with E, B with C. The tritone between them resolves inward — F falls to E and B rises to C — and those two arrival notes are the third and the root of the tonic. That is what is special about the major scale: a single pair of notes that points, unambiguously, home.
The tritone inside the dominant
The fifth-degree chord with a seventh contains that pair. G with a seventh has G, B, D and F: B and F are the scale's tritone, and it is because of them that the chord is dominant. Take the tritone out and what is left is G with D, a hollow fifth that asks for nothing.
Every dominant, in every key, carries the tritone between its third and its seventh, and it is that tritone that resolves. Two dominants sharing the same tritone resolve to the same place, which is the basis of tritone substitution.
Outside the key
The tritone also turns up where there is no dominant, and always with the same suspending effect. The diminished chord is made of them, two tritones crossed. The blues scale adds one to the pentatonic on purpose, to have a note that "asks". And the interval alone, with no chord, is the sound of an alarm, of a siren, of a question with no answer, used that way for centuries of music.
What to take from here
The tritone is six semitones, exactly half the octave, which is why it is the only interval whose inversion is itself. Written as an augmented fourth it opens; written as a diminished fifth it closes. There is only one in the major scale, between the fourth and seventh degrees, pointing at the tonic, and it is what makes the dominant dominant. Nearly all the tension in tonal harmony is this interval asking to move.