A song consists of two main components: a lyric and a piece of music. To understand a song, let's look at its two components in turn.
A lyric is a passage of text containing far less words than a short story or speech. It can document an event, tell a fictitious story, state an opinion, offer advice, or tell the listener anything else which can be expressed in words. Generally speaking, only two characteristics are required in a popular lyric:
Firstly, it should be emotionally engaging. The emotion can be any you like, positive or negative, but it has to be strong. It must engage the listener, resonating with him and inspiring him.
The other thing a good lyric needs to do is form patterns. That way, the listener can play a game — probably on the subconscious level — in which he tries to guess what the singer will say next. Don't make it too easy, of course: the game is only fun when it presents a challenge. Just like a good joke or plot twist, the craft lies in the ability to technically fulfill a promise — such as making a rhyme — but doing it in a way that the listener didn't see coming.
The methods used to form the patterns are formally known as meter, rhythm, rhyme, alliteration, assonance and consonance, among others, but discussing them is outside of the scope of this humble essay. What's important is to get the balance right between making the lyric too simple and too complex.
Next up is the music. Music consists of timbres, rhythms, harmonies and melodies. In pop songs, the lyric is generally sung in the main melody, meaning that the singer is doing two jobs at once: getting a point across with her words while also playing the part of the lead instrument.
The music should do the same two things as the lyric: it should engage the listener emotionally in the way that only music can, and it should form patterns. The more complex its patterns are, the worse it will sound at first because it is less predictable. After a few listens, however, it will sound better than a simpler song, which by that time will have become so easy to predict and memorise that it will have started to sound comparatively dull.
In summary, both the lyric and the piece of music are interesting to us because they fulfill both of our desires: to feel strong emotions, and to spot patterns. I suspect this is simply because we are essentially just emotionally charged machines that constantly try to analyse and predict patterns, but really, you'd be better off asking a biologist.