Siglo literally translates to "century". It's a powerful and dramatic word that forms the core of this song's emotional message.
In the title and chorus, the singer laments, "Un día es un siglo sin ti" (A day is a century without you). He uses this hyperbole to express how time feels impossibly slow and empty without his loved one, turning a single day of absence into an eternity of longing.
This song is one giant, dramatic apology set to an upbeat Argentinian cuarteto rhythm. The singer is desperate to win someone back and is trying everything he can think of, even inventing "Mil y una historia" [A thousand and one stories] just to be near them. He knows his ex thinks he's insincere and has no hope of changing, but he's here to prove them wrong.
The whole song is a plea for a second chance. He admits he made mistakes and is asking for forgiveness "De la única forma que sé" [In the only way I know how]: by leaving the door to his heart wide open. He insists this person made him grow and that he's "no soy el mismo de ayer" [not the same man as yesterday]. The feeling is summed up in the title, a classic over-the-top romantic line: "Un día es un siglo sin ti" [A day is a century without you].