Júrame means "swear to me" or "promise me." It's the imperative form of the verb jurar (to swear or to promise).
In this classic bolero, Julio Iglesias passionately implores his beloved to swear that she will never forget him or their love. It's a powerful and emotional plea that sets the tone for the entire song, making it a central and memorable word.
"Júrame" is a classic bolero about a love so sudden it catches the singer off guard. He admits that nobody has ever seen him fall this hard before, and even he can't fully explain what's happening to his own heart. He's a little jealous too, not wanting anyone else to ever cross her mind.
The song builds into a tender, pleading refrain: swear to me that you won't forget this moment, look at me, kiss me the way no one ever has, and love me until it hurts. It's a song about wanting proof of love, not because he doubts her, but because what he feels is so overwhelming he needs to hear it said out loud.