"Los sábados por la noche."
My understanding is that 'por la noche' [or 'en la noche'] refers to 'at night' or within the night at no set time, as 'por la mañana' would be 'in the morning' and 'por la tarde' would be 'in the afternoon'. So, 'los sábados por la noche' is 'Saturdays at night' or the given translation of 'Saturday nights'. Note that specific times take 'de la mañana' as in: 'son las 10:00 de la mañana'.
So many nitpickers! Its not a complete sentence! Its not 100% proper grammar! Nobody uses 100% correct grammar. And there is no way to learn a language 100% without being fully immersed. Hasn't anyone heard the term "lost in translation"? Learn the best you can, then start talking to people who really know the language to sharpen the rough edges.
IN English you could say 'On Saturday in the night', you can say 'sabado noche' if the context helps. is as 'sabado noche' is all noun. But if you are going to say 'this happens on saturday night' in spainish you need all the articles and prepositions, 'esto pasa los sabados por la noche'