Tatuajes literally translates to "tattoos." While it's a common word on its own, its use in this song is what makes it so powerful and memorable.
Junior H creates a beautiful metaphor, singing "Y tus besos son tatuajes, nunca se arrancan de mí" (And your kisses are tattoos, they are never torn from me). He uses the image of a permanent tattoo to describe how the memory of a past love is indelibly marked on him, an emotional pain that cannot be removed.
This song is a whole story about a breakup, but the best part is the conversation Junior H has with his ex-girlfriend's mom. It starts with him and his girlfriend in a tense moment. He feels something is wrong and knows it's over, bluntly saying he'll have to delete everything: "A la verga, se fueron nuestros recuerdos" (To hell with it, our memories are gone). He feels like he gave her everything, and her memory is now stuck to him like a tattoo.
Then, the song switches to his ex's mom confronting him. She saw them happy and now she sees him acting like a jerk, getting more tattoos, and calling himself a 'sad boy'. Her questions are so real: "que qué es eso de 'sad boy'" (what is that 'sad boy' thing?). His answer is the perfect comeback and the heart of the song. He tells her straight up, "¿Doña, qué hago si su hija me lastimó?" (Ma'am, what can I do if your daughter hurt me?). He's explaining that his whole new tough, sad persona is because of the heartbreak she caused.