Aurora literally translates to "dawn" or "daybreak." It's a beautiful and somewhat poetic word that isn't as common in everyday conversation as other time-related words.
In the song, the lyric "El tiempo no perdona y se va igual que la aurora" (Time does not forgive and leaves just like the dawn) uses aurora to evoke the fleeting nature of time. Just as the dawn quickly gives way to day, time passes swiftly and relentlessly, emphasizing the song's theme of cherishing moments before they are gone.
Alex Fernández's "El Tiempo No Perdona" is a really touching song about the bittersweet feeling of watching someone you love grow older. He sings about time slipping away "Poco a poco te me vas" [Little by little you are leaving me] and how he can't do anything to stop it. It's a universal feeling, that wish to just pause a moment with someone special.
He reflects on being a child and wanting to grow up to be like his dad, even asking God to speed up time. But now, he's changed his mind, wishing he could stop time instead. He pleads, "Detente, no camines hacia donde corre el tiempo" [Stop, don't walk where time runs], wanting to buy more time with his father, even offering his savings to do so. It's a beautiful expression of love and the desire to hold onto those precious moments.