Song of Songs is no ordinary story, so don't be tempted to look for a linear storyline with exposition and a dénouement and other such fancy words. Nope, this is a collection of love poetry. And much like Shakespeare's love sonnets, the poetry in Song of Songs doesn't go from point A to point Z.
But enough hemming and hawing: let's get down to what happens. So we have two lovers who can't get married because of some pesky familial and social barriers. They pine for each other like whoa and tell the audience—in some TMI detail—exactly how much they dig the other. In the end, we get an inkling that the lovers will eventually be married, but we're not sure that's entirely the point.