As in many folk songs and ballads, the song opens with a call for people to unite. However, the song does not want to tell a story about the past. Instead, it is a call to action that aims to make people aware that society is changing at a rapid pace and that these changes are irrevocable and cannot be ignored. Consequently, in order to survive, Dylan advises that we stop doing nothing and start adapting to the changes surrounding us ("If your time to you/ Is worth savin'/ Then you better start swimmin'/ Or you'll sink like a stone").
While the words are very moving on their own, there is something about the combination of the impassioned lyrics with the strong melody that makes listening to this song a moving experience. Musically, the melody does not hold any surprises - it remains steady without any surprising dips or key changes. Together, the lyrics and melody simultaneous create a feeling of urgency ("Come mothers and fathers/ Throughout the land/ And don't criticize/ What you can't understand") while maintaining a sense of inevitability ("Please get out of the new one/ If you can't lend your hand"). It is this dichotomy and balance that, to me, makes the song so successful. The real passion (and potential anger) that comes with the song is tempered by the utter certainty in the lyrics and music.
What I find fascinating about this song is the number of covers people have done of it. In addition to the Bob Dylan original, it seems like almost every artist in the past 25 or so years has covered it, which speaks to the song's continued relevance (its this or recording this song is a rite of passage for musicians). For instance, this song has been covered by Billy Joel, Bruce Springsteen, The Beach Boys, Simon & Garfunkel, Phil Collins, and Peter, Paul & Mary. If you are still scratching your head over Phil Collins and The Beach Boys doing this song, try to wrap your mind around the fact that Burl Ives, who is best-known for his role as Sam the Snowman in the stop-motion special Rudolph the Red-Nosed Reindeer, did a version of this song.
Despite the potential humor that listening to some of these covers might invite (just imagine Sam the Snowman whipping out his banjo and singing this song to Rudolph and Santa), for my money, if you want to go with a cover of "The Times They Are A-Changin,'" your best bet is Tracy Chapman's version: