A musical interval is the distance between two notes that are played. So intervals are always relative. E and A for example are a perfect fourth apart. In other words, there are four whole steps between them.
My crude understanding of intervals is that they act like the DNA of a particular song -- the intervals and chord progressions essentially determine its characteristics. Not only do they involve going from one note or cluster of notes to another, but they involve playing them at the same time as well. This is how chords acheive their characteristics. Playing a G, B and D together (which would be the 1st, 3rd, and 5th notes in the G major scale) is a strong and confident sounding chord -- this is a G major chord. However, if one of those notes is off by just a half step, you can get a completely different sound.
One of the most popular arrangements of intervals in pop music is the I, IV, V chord progression. If you pick up a guitar and can play the A, D and E major chords, you will be amazed out how many pop songs you can play or at least approximate with those chords. The next time you hear Eddie Cochran's classic "Summertime Blues," for example, just think of the guitar parts as I-IV, V-I, I-IV, V-I. The Who nailed the visceral nature of the I, IV, V chord progression when they remade this song. It's fair to say that the Ramones probably wouldn't exist if it weren't for this chord progression. Since a chord is generally a combination of at least three notes, and there are 12 notes out there, there are dozens of chord combinations that can result, and the difference in just one chord can make a person go from being happy to depressed.
Chord progressions and intervals have a spiral-like relationship -- there are numbered notes within chords, numbered chords within songs, and the rules apply across the board. I deal mostly with simple intervals and triad chords on this site, but there are a bunch of other intervals and chords in a myriad varieties that a musician can use, such as diminished, suspended, augmented, dominant, and so on. All of these arrangements of sounds create effects that are extremely powerful, a suspended chord for example really does leave the listner hanging.
There isn't a lot of sinnuendo in music theory -- it's a pretty bland thing at the theoretical level. But there is one interval that is of especial interest, and that's the "Devil's interval."
This is the tritone -- two notes that are an augmented fourth or a diminished fifth apart. During medieval times it was forbidden to play the tritone, it really was considered a tool of the Devil because it sounded so incredibly dissonant. The effect it produces is shocking -- it creates an unmistakably disturbing vibe when it's heard. Danny Elfman used the tritone when he wrote the theme to "the Simpsons," If you listen to the opening of the song, it's easy to see what he was gunning for. To me, his composition has an impish quality to it; it evokes mischief and gives off the impression that everything isn't as it seems in Springfield. Occasionally you'll hear the Devil's interval in pop music, but it's almost always to produce that bizarre dissonant quality. The Hendrix song "Purple Haze" opens with tritones, as does the Rush song "YYZ." The intros to both tunes are intentionally designed to shock the listener, to create uneasiness.