Tatuajes means "tattoos" in Spanish. It's an evocative word that immediately brings to mind permanence and deep personal meaning.
In this song, Edgardo Nunez uses "tatuajes de tus besos" (tattoos of your kisses) as a powerful metaphor. He's not talking about literal tattoos, but rather the indelible mark left by a past love, suggesting that the memories and feelings are so profound they're etched onto his very being, just like a tattoo.
Edgardo Núñez spends "Tatuajes" trying to talk himself into a breakup he doesn't actually want. He opens with "Hiriendo mi felicidad" ("Hurting my happiness") as his reason for going, then undercuts it two lines later by admitting "Aunque eres mi necesidad" ("Though you're my need"). By the end of the verse he can't even commit to getting over her: "Pero eso de que te olvide" ("But as for forgetting you"), and he leaves the sentence hanging with no answer.
The chorus is where the title shows up. "Tatuajes de tus besos llevo en todo mi cuerpo" ("I wear tattoos of your kisses all over my body") turns the memory into something permanent and unchosen, and he pushes the image further by calling it a habit he can't kick: "Se me hizo vicio, ver tus ojos, respirar tú aliento" ("Seeing your eyes, breathing your breath became my addiction"). The second verse has him claiming to move on with "Con otro amor el corazón" ("With another love in my heart"), but the song closes exactly the way it opened, with him still carrying her around inside. The new love sounds more like a patch job than a fix.