Siglo literally means "century". It's a great example of a hyperbole, a figure of speech that uses extreme exaggeration to make a point.
In this passionate ballad, Chayanne sings "Un día es un siglo sin ti" (A day is a century without you). He uses this dramatic comparison to express the unbearable and seemingly endless pain of being separated from the one he loves. This powerful metaphor is what makes the song so memorable.
Ever felt like time slows down when you miss someone? Chayanne takes that feeling and turns it up to eleven. In "Un Siglo Sin Ti," he declares that "Un día es un siglo sin ti" (One day is a century without you). This is a song about massive regret and a desperate plea for a second chance. The singer knows his ex thinks he's insincere and a lost cause, but he's realized he can't live without her.
He's not just saying sorry; he's trying to prove he's a new man. He admits, "Si te he fallado / Te pido perdón" (If I have failed you / I ask for your forgiveness) and makes a grand, romantic gesture: he's "Abriendo las puertas de mi corazón / Para cuando decidas volver" (Opening the doors of my heart / For when you decide to return). He credits her with completely changing him for the better, insisting "no soy el mismo de ayer" (I'm not the same as I was yesterday). It's a huge, heartfelt apology from someone who has learned his lesson the hard way.